IODS course project
Richard E. Lamprecht
- About the project
- Chapter 2: Regression and model validation
- Chapter 3: Logistic regression
- 1. Data wrangling
- 2. Analysis
- 2.1. Read the prepared data set
- 2.2. The data set
- 2.3. Performing the analyses
- 2.3.1. Choose variables for the analyses
- 2.3.2. Explore the distributions of the chosen variables numerically and graphically
- 2.3.3. Logistic regression - explore the relationship between the choosen variables and the high alcohol consumption
- Bonus tasks
- Chapter 4: Clustering and classification
- 1. Analysis of Boston data set
- 1.1. Load the data set & check the structure
- 1.2. The data set
- 1.3. Data set analysis
- 1.3.1. Data standardization (scaling of the data set)
- 1.3.2. Create a categorical variable (crime rate)
- 1.3.3. Define training and test data set
- 1.3.4. Linear discriminant analysis of the training data set
- 1.3.5. Prediction of crime classes on the test data set
- 1.3.5. Reload and standardize the Boston dataset and calculate the distances
- 1.3.5. Determine the k
- 1.3.6. Bonus
- 1.3.7. Super-Bonus
- Chapter 5: Dimensionality reduction techniques
- Chapter 6: Analysis of longitudinal data
- 1. Load and check the data sets
- 1.1. Load the wide (bprs & rats) and long (bprs_l & rats_l) data sets
- 1.2.The long data sets (bprs_l & rats_l)
- 2. Analysis of the long data sets (bprs_l & rats_l)
- 3. Analyses of chapter 8 of MABS using the RATS data (rats_l)
- 4. Analyses of Chapter 9 of MABS using the BPRS data (bprs_l)
About the project
The course:
The course Introduction to open data science started on Wednesday 30th October 2019.
My feelings:
I’m feeling so excited!!
I got aware of the course by an announcement some weeks ago on the University of Eastern Finland’s yammer platform. After checking the course material on MOOC platform I see great new things coming up for me to learn. So I’m very excited about it.
I’m looking forward to meet all the new things I will learn in this course and especially I’m looking forward to meet all challenges coming up in this course.
I hope I can manage all assignments.
I’m expecting to learn a lot about data handling, improve my R skills and hopefully be able to use this knowledge later on in my work.
Links to my new GitHub pages:
Richard’s GitHub IODS repository
I’m trying to keep everything up to date!
Chapter 2: Regression and model validation
Data wrangling and regression analysis
Work of week 45 (4.11. - 10.11.2019)
1. Data wrangling
4.11.2019: Started to work through the DataCamp exercises offered by the IODS course and was able to finish
- R Short and Sweet
- R Helsinki Open Data Science courses
5.11.2019: Start to work on R script for data wrangling and regression analysis
- Wrote the script for data wrangling and was able to finish that part (all tests on the script worked)
Now follows a discription through my work progress.
1.1. Read the dataframe
Script of the data read (reading the dataframe from the website)
# Read the data file ----
learning2014 <- read.table(file =
"https://www.mv.helsinki.fi/home/kvehkala/JYTmooc/JYTOPKYS3-data.txt",
sep = "\t", header = TRUE)
# read the data file and assign it to the object "learning2014"
# Explore the read data ----
head(learning2014) # see the top of the dataset with the first 6 observations
str(learning2014) # check the structure of the dataset --> --> (observations = rows and variables = columns)
dim(learning2014) # check the table dimensions --> (observations = rows and variables = columns)The dataset consists of 183 observations (rows) in 60 variables (columns).
If you want to check the data also with some diagrams, you can use for example following plots:
# Structure of some the data in plot or in a histogram (here the variable Points from)
library(ggplot2)
qplot(Attitude, Points, data = learning2014)
hist(learning2014$Points)What follows now is the data wrangling. We have to summarize variables of deep, surface and strategic learning approaches and calclulate the mean value of the summarized variables. Other variables stay (gender, Attitude, Points), only the column names are changed.
1.2. Perform the data wrangling
Script for data wrangling
I decided to do the data wrangling into a new R object (a dataframe) which I named “lrn14_analysis”. This dataframe will later be used for the data analyses.
Here I used a more complicated way to do the data wrangling, in the meeting on Wednesday, 6.11. a way with the pipe operator %>% was presented.
# Create an analysis dataset: "lrn14_analysis" ----
# Create an analysis dataset with the variables gender, age, attitude, deep, stra, surf and points
# (by combining questions in the learning2014 data)
library(dplyr) # load the package for data wrangling
keep_columns <- c("gender","Age","Attitude","Points") # these are the data columns which need to be kept
lrn14_analysis <- select(learning2014, one_of(keep_columns)) # assin a new object and select kept columns
colnames(lrn14_analysis) <- c("gender", "age", "attitude", "points") # change of the kept column names
# define questions (observations from variables) acc. instructions
deep_q <- c("D03", "D11", "D19", "D27", "D07", "D14", "D22", "D30","D06", "D15", "D23", "D31") # deep questions
surf_q <- c("SU02","SU10","SU18","SU26", "SU05","SU13","SU21","SU29","SU08","SU16","SU24","SU32") # surface questions
stra_q <- c("ST01","ST09","ST17","ST25","ST04","ST12","ST20","ST28") # strategic questions
# Select the combined variables (columns) & scale the observations (mean) and add it to the analysis dataframe
deep <- select(learning2014, one_of(deep_q))
lrn14_analysis$deep <- round(rowMeans(deep, na.rm = TRUE), digits = 2) # values are rounded to 2 digits
surf <- select(learning2014, one_of(surf_q))
lrn14_analysis$surf <- round(rowMeans(surf,na.rm = TRUE), digits = 2)
stra <- select(learning2014, one_of(stra_q))
lrn14_analysis$stra <- round(rowMeans(stra, na.rm = TRUE), digits = 2)
# devide the number of the attitude column by 10 (its a sum of 10 questions)
lrn14_analysis$attitude <- lrn14_analysis$attitude / 10
# Exclude observations where the exam points variable is zero.
lrn14_analysis <- filter(lrn14_analysis, points > 0)The new dataframe was created. Now the structure is checked.
# Check the analysis dataset
str(lrn14_analysis)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 4.75 3.83 3.25 4.33 4 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 2.42 1.92 2.83 2.17 3 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 3.62 2.25 4 4.25 3.5 ...dim(lrn14_analysis)## [1] 166 7The data consists now of 7 variables (columns) and 166 observations. The object name is “lrn14_analysis”.
1.3. Safe the updated dataframe as a .txt table or .csv table
First the the working directory is set and then I save the dataframe with write.table() and the write.csv()functions.
# Set the working directory to IODS project folder ----
setwd("~/IODS-project") # set the wd to the IODS folder# safe the analysis dataset to the "data" folder ----
write.table(lrn14_analysis,
file = "C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/lrn14_analysis_table.txt",
sep = "\t", col.names = TRUE, row.names = TRUE)
write.csv(lrn14_analysis,
file = "C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/lrn14_analysis_table.csv",
row.names = FALSE)Afterwards I check if the tables can be read by R using the read.table() and read.csv() function plus the %>%operator piping to the head() function showing the first 6 observations of the dataframes.
# check if the table can be read ----
read.table(file =
"C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/lrn14_analysis_table.txt") %>% head()## gender age attitude points deep surf stra
## 1 F 53 3.7 25 3.58 2.58 3.38
## 2 M 55 3.1 12 2.92 3.17 2.75
## 3 F 49 2.5 24 3.50 2.25 3.62
## 4 M 53 3.5 10 3.50 2.25 3.12
## 5 M 49 3.7 22 3.67 2.83 3.62
## 6 F 38 3.8 21 4.75 2.42 3.62read.csv(file =
"C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/lrn14_analysis_table.csv") %>%head()## gender age attitude points deep surf stra
## 1 F 53 3.7 25 3.58 2.58 3.38
## 2 M 55 3.1 12 2.92 3.17 2.75
## 3 F 49 2.5 24 3.50 2.25 3.62
## 4 M 53 3.5 10 3.50 2.25 3.12
## 5 M 49 3.7 22 3.67 2.83 3.62
## 6 F 38 3.8 21 4.75 2.42 3.622. Analysis
The work on the analysis script and documentation started on 6.11.2019.
2.1. Reading the data
Read the dataset table from my data folder and checked the dataframe structure and dimensions.
# Set the working directory
setwd("~/IODS-project/data") # set work directory# Read the data file ----
lrn14_analysis <-
read.table(file = "C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/lrn14_analysis_table.txt", stringsAsFactors = TRUE)
lrn14_analysis %>% str() # read the data table and check the structure## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 4.75 3.83 3.25 4.33 4 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 2.42 1.92 2.83 2.17 3 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 3.62 2.25 4 4.25 3.5 ...When I read the data table I set the “stringsAsFactor” agument (in the read.table function) to TRUE. So the observations in the gender column “F” and “M” will become factors - here 1 & 2. The dataframe includes 166 observations (the rows) in 7 variables (the columns).
2.2. Graphical overview and data summary
The data we analyise consists of a survey from students relating their learning approaches. The data includes the students’ gender, age, exam points and the global attitude towards statistics (consisting of a sum of 10 questions related to students attitude towards statistics, each measured on the Likert scale (1-5). The attitude value was divided by 10 in the data wrangling part to show the value in the 1-5 scale.
For a graphical overview of the dataset I used the ggpairs() function which results in several plots and correlations of the observations between the different variables.
library(ggplot2)
library(GGally) # to show the graph these packages need to be loaded
ov_lrn14_2 <- ggpairs(lrn14_analysis, mapping = aes(col = gender), title = "Graphical overview of lrn14_analysis",
lower = list(combo = wrap("facethist", bins = 20)),
upper = list(continuous = wrap("cor", size = 2.8)))
ov_lrn14_2 # show the graph
The overview plot shows the data distribution of all observations of each variable with a histogramm, Q-plots (point diagrams) and the line graphs. The data colours represent the different gender of the students of the data frame. Here female students are shown in redish colour and male students in turquise colour.
In the upper right part of the overview graph the data correlations (all observations of one variable correlating to the observations of another varibable) is shown. These results give a first hint which variable might show a significant regression analysis result.
# Save the overview plot
ggsave("OV_plot_lrn14.png",
plot = ov_lrn14_2, path = "~/IODS-project/data/", scale = 1, dpi = 300)
# the graph is saved as .png file in my data folderAdditionally a data summary table of the lrn14_analysis dataset was created.
# Summary table of the lrn14_analysis data ----
Sum_table <- summary(lrn14_analysis)
Sum_table## gender age attitude points deep
## F:110 Min. :17.00 Min. :1.400 Min. : 7.00 Min. :1.58
## M: 56 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:19.00 1st Qu.:3.33
## Median :22.00 Median :3.200 Median :23.00 Median :3.67
## Mean :25.51 Mean :3.143 Mean :22.72 Mean :3.68
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:27.75 3rd Qu.:4.08
## Max. :55.00 Max. :5.000 Max. :33.00 Max. :4.92
## surf stra
## Min. :1.580 Min. :1.250
## 1st Qu.:2.420 1st Qu.:2.620
## Median :2.830 Median :3.185
## Mean :2.787 Mean :3.121
## 3rd Qu.:3.170 3rd Qu.:3.620
## Max. :4.330 Max. :5.0002.3. Regression models
2.3.1. Simple regression model of point ~ attitude
I was testing different simple regression models with the exam points as dependent variable (y) using function lm().
# Simple regressions ----
# regression with gender variable
gen_lm <- lm(points ~ gender, data = lrn14_analysis)
summary(gen_lm) # not significant
# regression with age variable
age_lm <- lm(points ~ age, data = lrn14_analysis)
summary(age_lm) # not significantThese regression analysis did not result in significant outcomes. Next the I performed the regression analysis of points ~ attitude. This had following outcome:
# attitude
att_lm <- lm(points ~ attitude, data = lrn14_analysis)
att_lm_res <- summary(att_lm)
att_lm_res##
## Call:
## lm(formula = points ~ attitude, data = lrn14_analysis)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.6372 1.8303 6.358 1.95e-09 ***
## attitude 3.5255 0.5674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09knitr::kable(att_lm_res$coefficients, digits=3, caption="Regression coefficients point ~ attitude")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 11.637 | 1.830 | 6.358 | 0 |
| attitude | 3.525 | 0.567 | 6.214 | 0 |
Here the diagnostic plots of the regression model points ~ attitude:
plot(att_lm, which = c(1,2,5))
Regression model point ~ attitude interpretation:
Significant outcome was shown in the simple regression analysis of point ~ attitude.
The summary of the regression model shows that the estimate increase of points by a value of 1 of attitude is 3.525. The p-value of 4.12e-09 *** represents that as asignificant result (it’s pretty much 0). So the students’ attitude (relating statistics) has a significant influence in the exam points outcome. So depending on the attitude the student has a higher number of points or a lower one. But the intercept shows a value of 11.63 - so even with zero attitude a student would still reach a level of 11 points at an exam.
The multiple R-squared represents the correlation coefficient. It is the square root of R-squared (the coefficient of determination). The multiple R-squared explains how strong the linear relationship is, which would be strongest with a value of 1 and weakest with a value of 0 (no relationship). Here the value is 0.1906, so not a very strong relationship, but it has an effect.
The diagnostic plots consist of a "Residual vs. Fitted plot which is used to check the linearity (or non-linearity) of the observations, eventual variance errors (observations which concentrate in a certain direction) and you can also check if there are possible outliers in your dataset.
The resulting plots of our points ~ attitude model show a good linearity and a good variance. The observations are nicely distributed over the base line and you cannot see any direction were the points might show a variance difference. There are also three outliers (145, 56, 35), but these are not disturbing the analysis and don’t have to be removed since they are still centered in the overall data point distribution.
The Normal Q-Q plot helps us to check the data distribution of our dataset (normal distributed or exponential). Our datapoints are nicely distributed over the line and are normal distributed. That means that our analysis does not need any transformed data points before any further statistical analysis (for example logarithm or square root transformation).
The Residual vs. Leverage plot helps us to identify if there are observations which might have a high impact on a statistical model. So the observations in our case show very low leverage and even the points which are further away from the others show low leverage and so cannot be considered as outliers.
Additionally here is a plot representing points vs. attitude with the regressions line. For me it makes it easier to understand the whole thing when the data is shown is a graph.
library(ggplot2)
qplot(attitude, points, data = lrn14_analysis) + geom_smooth(method = "lm")
In the graph someone can check if the estimate increase of 3.525 per one unit of attitude is shown by the regression line.
2.3.2. Multiple regression model of point ~ attitude + stra
Here I’m checking the linear regression of points ~ attitude and strategic learning approach outcome - I want to know if the strategic learning approach has a relationship to the exampoints of the students.
# points ~ attitude + stra ----
att_st_lm <- lm(points ~ attitude + stra, data = lrn14_analysis)
att_st_lm_res <- summary(att_st_lm)
att_st_lm_res##
## Call:
## lm(formula = points ~ attitude + stra, data = lrn14_analysis)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.6482 -3.3135 0.5571 3.7966 10.9300
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.9725 2.3966 3.744 0.000251 ***
## attitude 3.4664 0.5652 6.134 6.27e-09 ***
## stra 0.9132 0.5345 1.709 0.089438 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.289 on 163 degrees of freedom
## Multiple R-squared: 0.2048, Adjusted R-squared: 0.195
## F-statistic: 20.99 on 2 and 163 DF, p-value: 7.746e-09knitr::kable(att_st_lm_res$coefficients, digits=3, caption="Regression coefficients point ~ attitude + stra")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 8.972 | 2.397 | 3.744 | 0.000 |
| attitude | 3.466 | 0.565 | 6.134 | 0.000 |
| stra | 0.913 | 0.534 | 1.709 | 0.089 |
Here the diagnostic plots of the regression model points ~ attitude + stra:
plot(att_st_lm, which = c(1,2,5))
Regression model point ~ attitude + stra interpretation:
In this multiple regression I wanted to see if the strategic learning approach shows an influence on the amount of exampoints a student can reach. The outcome shows that the strategic learning approach has some influence but not a very strong one. Also the p-value of 0.08 shows that the significance of the influence is not very strong. The multiple R-squared is a bit higher compared to points ~ attitude regression model so we have a little higher correlations if we put the strategic learning approach into account.
The diagnostic plots show more or less the same results as in the regression model point ~ attitude. Data is normal distributed and no outliers are significantly disturbing the analysis.
2.3.3. Multiple regression model of point ~ deep + surf + stra
In the third regression model I put points vs. deep, surface and strategic learning approach into account. Does the combination of deep, surface and strategic learning approach have an influence in the exampoints of students.
# points ~ + deep + surf + stra ----
de_su_st_lm <- lm(points ~ deep + surf + stra, data = lrn14_analysis)
de_su_st_lm_res <- summary(de_su_st_lm)
de_su_st_lm_res##
## Call:
## lm(formula = points ~ deep + surf + stra, data = lrn14_analysis)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.1208 -3.0725 0.5196 4.2798 10.3346
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.9426 5.1147 5.268 4.34e-07 ***
## deep -0.7472 0.8659 -0.863 0.3895
## surf -1.6328 0.9149 -1.785 0.0762 .
## stra 0.9850 0.5962 1.652 0.1005
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.827 on 162 degrees of freedom
## Multiple R-squared: 0.04072, Adjusted R-squared: 0.02296
## F-statistic: 2.292 on 3 and 162 DF, p-value: 0.0801knitr::kable(de_su_st_lm_res$coefficients, digits=3, caption="Regression coefficients point ~ deep + surf + stra")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 26.943 | 5.115 | 5.268 | 0.000 |
| deep | -0.747 | 0.866 | -0.863 | 0.389 |
| surf | -1.633 | 0.915 | -1.785 | 0.076 |
| stra | 0.985 | 0.596 | 1.652 | 0.100 |
Here the diagnostic plots of the regression model points ~ + deep + surf + stra:
plot(de_su_st_lm, which = c(1,2,5))
Regression model point ~ deep + surf + stra interpretation:
The outcome of this regression model shows that with a minor significance (0.0762) the surface learning approach has some indluence to the exam outcomes of students. Checking the estimate for surf show a negative value of -1.63 which means that students with a surface learning approach tent to have 1.6 points less as a test result with a increased surface learning approach value of 1. The deep learning approach seem to have no influence at all in the amount of points students reach at the exams and the stratetic approach has a non-significant positive influence.
The multiple R-squared of 0.04 is very low and explains that there is more or less no correlation between these variables and the exampoints from the students.
Here I put the three variables into a qplot() to show the relationsships of the learning approaches and the exampoints.
library(ggplot2)
par(mfrow = c(2,2))
qplot(deep, points, data = lrn14_analysis) + geom_smooth(method = "lm") + ggtitle("Points vs. deep learning approach")
qplot(surf, points, data = lrn14_analysis) + geom_smooth(method = "lm")+ ggtitle("Points vs. surface learning approach")
qplot(stra, points, data = lrn14_analysis) + geom_smooth(method = "lm")+ ggtitle("Points vs. strategic learning approach")
So, this was a trial to analyse the dataset with simple and multiple regression models. I hope the interpretations are easily understood and the diary structure is not confusing you.
I had fun preparing this first statistics and graph chapter and I’m looking forward to learn more…
Chapter 3: Logistic regression
Data wrangling and performing a logistic regression analysis
Work of week 46 (11.11. - 17.11.2019)
2. Analysis
2.1. Read the prepared data set
The working directory is set using setwd() and the data file “alc” prepared in the data wrangling part is read using read.table()function. Afterwards the data frame is checked.
alc <- read.table(file =
"C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/alc_table.txt", stringsAsFactors = TRUE)
# read the data file and leave the binary values as they are
# Check the data frame
dim(alc)## [1] 382 35str(alc)## 'data.frame': 382 obs. of 35 variables:
## $ school : Factor w/ 2 levels "GP","MS": 1 1 1 1 1 1 1 1 1 1 ...
## $ sex : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
## $ age : int 18 17 15 15 16 16 16 17 15 15 ...
## $ address : Factor w/ 2 levels "R","U": 2 2 2 2 2 2 2 2 2 2 ...
## $ famsize : Factor w/ 2 levels "GT3","LE3": 1 1 2 1 1 2 2 1 2 1 ...
## $ Pstatus : Factor w/ 2 levels "A","T": 1 2 2 2 2 2 2 1 1 2 ...
## $ Medu : int 4 1 1 4 3 4 2 4 3 3 ...
## $ Fedu : int 4 1 1 2 3 3 2 4 2 4 ...
## $ Mjob : Factor w/ 5 levels "at_home","health",..: 1 1 1 2 3 4 3 3 4 3 ...
## $ Fjob : Factor w/ 5 levels "at_home","health",..: 5 3 3 4 3 3 3 5 3 3 ...
## $ reason : Factor w/ 4 levels "course","home",..: 1 1 3 2 2 4 2 2 2 2 ...
## $ nursery : Factor w/ 2 levels "no","yes": 2 1 2 2 2 2 2 2 2 2 ...
## $ internet : Factor w/ 2 levels "no","yes": 1 2 2 2 1 2 2 1 2 2 ...
## $ guardian : Factor w/ 3 levels "father","mother",..: 2 1 2 2 1 2 2 2 2 2 ...
## $ traveltime: int 2 1 1 1 1 1 1 2 1 1 ...
## $ studytime : int 2 2 2 3 2 2 2 2 2 2 ...
## $ failures : int 0 0 2 0 0 0 0 0 0 0 ...
## $ schoolsup : Factor w/ 2 levels "no","yes": 2 1 2 1 1 1 1 2 1 1 ...
## $ famsup : Factor w/ 2 levels "no","yes": 1 2 1 2 2 2 1 2 2 2 ...
## $ paid : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 2 ...
## $ activities: Factor w/ 2 levels "no","yes": 1 1 1 2 1 2 1 1 1 2 ...
## $ higher : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
## $ romantic : Factor w/ 2 levels "no","yes": 1 1 1 2 1 1 1 1 1 1 ...
## $ famrel : int 4 5 4 3 4 5 4 4 4 5 ...
## $ freetime : int 3 3 3 2 3 4 4 1 2 5 ...
## $ goout : int 4 3 2 2 2 2 4 4 2 1 ...
## $ Dalc : int 1 1 2 1 1 1 1 1 1 1 ...
## $ Walc : int 1 1 3 1 2 2 1 1 1 1 ...
## $ health : int 3 3 3 5 5 5 3 1 1 5 ...
## $ absences : int 5 3 8 1 2 8 0 4 0 0 ...
## $ G1 : int 2 7 10 14 8 14 12 8 16 13 ...
## $ G2 : int 8 8 10 14 12 14 12 9 17 14 ...
## $ G3 : int 8 8 11 14 12 14 12 10 18 14 ...
## $ alc_use : num 1 1 2.5 1 1.5 1.5 1 1 1 1 ...
## $ high_use : logi FALSE FALSE TRUE FALSE FALSE FALSE ...colnames(alc)## [1] "school" "sex" "age" "address" "famsize"
## [6] "Pstatus" "Medu" "Fedu" "Mjob" "Fjob"
## [11] "reason" "nursery" "internet" "guardian" "traveltime"
## [16] "studytime" "failures" "schoolsup" "famsup" "paid"
## [21] "activities" "higher" "romantic" "famrel" "freetime"
## [26] "goout" "Dalc" "Walc" "health" "absences"
## [31] "G1" "G2" "G3" "alc_use" "high_use"The data frame consists of 35 variables with 382 observations.
2.2. The data set
The provided data is from a approach on student achievement in secondary education in two Portugal schools.
It was collected with school reports and questionaires and it includes:
- student grades
- demographic features
- social features
- school related features
The data consists of two data sets regarding the students’ performance in Mathematics (mat) and Portuguese language (por). In a publication by Cortez & Silva, 2008 this data was already used.
In the data wrangling part of this exercise the two data sets of mat and por were merged into one data set named “alc”. Columns (variables) which were not used in merging the data set were combined by averaging. Since we are interested to analyse the alcohol consumption of the students we created two new variables in the data set:
- “alc_use” –> calculated average of weekday (“Dalc”) and weekend (“Walc”) alcohol consumption
- “high_use” –> logical values created as TRUE or FALSE depending if the “alc_use” of a student is higher than 2
All other variables are explained in detail here (check attribute information).
2.3. Performing the analyses
2.3.1. Choose variables for the analyses
The task of this exercise is to figure out the relationships between high and low alcohol consumption and other variables in this data set. I have choosen to run the analyses on following four variables:
- sex –> student’s sex (binary: ‘F’ - female or ‘M’ - male)
- failures –> number of past class failures (numeric: n if 1<=n<3, else 4)
- famrel –> quality of family relationships (numeric: from 1 - very bad to 5 - excellent)
- absences –> number of school absences (numeric: from 0 to 93)
I have choosen these four variables for the data analyses for following reasons:
The student’s gender (“sex”) might have kind a great influence on the alcohol consumption. Young men tent to drink more alcohol for several reasons, women are able to control themselves (don’t take my steriotypical expressions serious please!).
The number of class failures (“failures”) of a student. Someone who fails in school more often might have a higher alcohol consumption then others (personal opionion) both as a causation of higher use or as a response to failure.
I think the overall quality of family relationships (“famrel”) might have a huge effect on a students alcohol consumption. I know, this is a very, very conservative point of view but in a country like Portugal (catholic, maybe more patriachic family structures) that might have an influence in young peoples alcohol consumption.
The number of school absences. I think if someone is absent from school they maybe spent their time also with drinking alcohol. I know there are other things to do as well, but when I was that age drinking beer was somehow a thing to spent your time.
2.3.2. Explore the distributions of the chosen variables numerically and graphically
The task is to explore the distribution of the variables I have choosen and the relationship with alcohol consumption numerically and graphically.
library(tidyr)
library(dplyr)
library(ggplot2)
library(knitr)
library(kableExtra)First I check how many female and male students are there:
# check the number of female and male students
alc %>% group_by(sex) %>% summarise(count = n()) -> fm
knitr::kable(fm, caption="Students") %>% kable_styling(bootstrap_options = "hover", full_width = FALSE, position = "center")| sex | count |
|---|---|
| F | 198 |
| M | 184 |
2.3.2.1. General data overview
This bar plot gives us an overview over the different variables and might give a hint which variables have a strong relationship with alcohol consumption.
2.3.2.2. General alcohol consumption overview
The graph shows the average alcohol consumption count of the students. 1 is the lowest consumption, 5 the highest consumption. The amount of students with a low consumption is around 130 of 385.
2.3.2.3. High alcohol consumption of students by gender
This graph gives an overview of the alcohol consumption of students devided by gender. With this graph you can see the amount of male or female students with higher alcohol consumption easier. High alcohol consumption (> 2) is indicated with turquise colour, lower alcohol consumption is indicated in red colour. You can see that the amount of male students with a high alcohol consumption is higher compared to female students.
2.3.2.4. High alcohol consumption and student’s class failures
What is seen in this graph is that students with higher alcohol consumptuion tent to have a higher class failing rate. Female students with higher alcohol use have a higher count at 1 failure, male students have a higher count at 2 failures or 3 failures.
Generally, you see that students which have no high alcohol consumption fail less in classes.
2.3.2.5. High alcohol consumption and student’s family relationship status
There are more students with a low alcohol conumption where the family relationship is good (4-5). Interestingly the amount of students with high alcohol consumption is also increasing with the quality of the family relationship status (higher count at status 3-5).
2.3.2.6. High alcohol consumption and student school abcences
High alcohol use seems to have an influence on the rate of school absences. Students with a higher alcohol consumption have an increased absance amount. Females tent to be more absent than males, also if they don’t fall in the group of high alcohol consumers.
2.3.3. Logistic regression - explore the relationship between the choosen variables and the high alcohol consumption
2.3.3.1. Calculation of the logistic regression model
I’m using in the model the four variables I have choosen: sex, failures, famrel and absences
# cv_glm, choosen variables logistic model
cv_glm <- glm(high_use ~ sex + failures + famrel + absences, data = alc, family = "binomial")
# Summary of the model (cv_glm_sum)
cv_glm_sum <- summary(cv_glm)##
## Call:
## glm(formula = high_use ~ sex + failures + famrel + absences,
## family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.1174 -0.8376 -0.5867 1.0091 2.1557
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.83877 0.53446 -1.569 0.1166
## sexM 0.99120 0.24540 4.039 5.37e-05 ***
## failures 0.42458 0.18854 2.252 0.0243 *
## famrel -0.27632 0.12837 -2.153 0.0314 *
## absences 0.09052 0.02252 4.020 5.83e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 419.78 on 377 degrees of freedom
## AIC: 429.78
##
## Number of Fisher Scoring iterations: 4| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | -0.839 | 0.534 | -1.569 | 0.117 |
| sexM | 0.991 | 0.245 | 4.039 | 0.000 |
| failures | 0.425 | 0.189 | 2.252 | 0.024 |
| famrel | -0.276 | 0.128 | -2.153 | 0.031 |
| absences | 0.091 | 0.023 | 4.020 | 0.000 |
The model output shows nice results and also significant one. So which variables influance high alcohol consumption?
- Being a male student significantly (p = 5.37e-05 ***) increases the probability of high alcohol consumption by ~1.
- Class failure have a smaller probability but still have a positive effect on high consumption (p = 0.0243 *).
- The quality of family relationships has a negative effect on high alcohol consumption (p = 0.0314 *).
- School absences show also a significant positive effect on high alcohol consumption (5.83e-05 ***).
This is a first result. But we are not going to use these results as our final interpretation of the logistic regression. We are checking now the odds ratios of the model.
2.3.3.2. Calculating the odds ratios
| odds ratios | 2.5 % | 97.5 % | |
|---|---|---|---|
| (Intercept) | 0.432 | 0.149 | 1.220 |
| sexM | 2.694 | 1.676 | 4.395 |
| failures | 1.529 | 1.057 | 2.228 |
| famrel | 0.759 | 0.589 | 0.976 |
| absences | 1.095 | 1.050 | 1.147 |
Interpretation of odds ratios: If the odds ratio is lower than 1 the risk of high alcohol consumption is lower, if it is greater than 1 the risk of high consumption increases. But we also have to have a look on the confidential interals - if the values cross the value of 1 the odds ratio cannot be significant! The odds ratios of this logistic model lead to following interpretation.
- sexM has a value of 2.7, the confidence interval is between 1.6 and 4.4. So this oods ratio tells us that the male students have a 2.7 times higher probability of consuming alcohol on a higher level.
- failures has a value of 1.5, the confidence interval is between 1.1 and 2.2. Failing in classes also increases the risk of high alcohol consumption by 1.5. So failing once increases the high alcohol consumption probability by 1.5.
- famrel has a value of 0.8, the confidence interval is between 0.6 and 1. It seems that the family relationships have a negative effect on high alcohol consumption.
- absences has a value of 1.095, the confidence interval is between 1.050 and 1.147. The amount of school absences increase the probability of higher alcohol consumption.
2.3.3.3. Explore the predictive power of the model
I calculated the probability of high alcohol use and added the data to the “alc” data frame. Then the prediction of a high alcohol consumption is defined to be a probability greater than 0.5 and also added this outcome to the “alc” data frame.
Here are the last 20 data points with the probability and prediction outputs:
## age Pstatus failures famrel high_use probability prediction
## 363 17 T 0 3 FALSE 0.28017549 FALSE
## 364 17 T 0 3 FALSE 0.29878836 FALSE
## 365 18 T 0 5 FALSE 0.09793643 FALSE
## 366 18 T 0 4 FALSE 0.15809170 FALSE
## 367 18 T 0 5 TRUE 0.11513474 FALSE
## 368 18 T 0 4 FALSE 0.12520393 FALSE
## 369 17 T 0 4 TRUE 0.33697399 FALSE
## 370 18 T 0 3 TRUE 0.42202546 FALSE
## 371 18 T 0 4 FALSE 0.31608605 FALSE
## 372 17 T 0 4 FALSE 0.33597108 FALSE
## 373 19 T 1 4 FALSE 0.37092103 FALSE
## 374 18 T 1 5 TRUE 0.45736187 FALSE
## 375 18 T 0 5 FALSE 0.10622930 FALSE
## 376 18 T 0 4 FALSE 0.19766624 FALSE
## 377 19 T 1 5 FALSE 0.16593043 FALSE
## 378 18 T 0 4 FALSE 0.14641340 FALSE
## 379 18 T 2 2 FALSE 0.41066768 FALSE
## 380 18 T 0 1 FALSE 0.30079024 FALSE
## 381 17 T 0 2 TRUE 0.49046497 FALSE
## 382 18 T 0 4 TRUE 0.31608605 FALSE| FALSE | TRUE | |
|---|---|---|
| High use | ||
| FALSE | 253 | 15 |
| TRUE | 82 | 32 |
Interpretation of the prediction:
The 2 x 2 cross table gives us the results of the model prediction for high alcohol consumption compared to the real numbers.
The model resulted in 253 correct false results and 82 false negative results. This means that the prediction fails to be correct. In 82 cases the model predicted that the student has no high alcohol consumption, but the student actually has high alcohol consumption. This false negative results should be much smaller.
In 15 cases the model predicts high alcohol consumption but is wrong, that’s a false positive prediction. 253 cases were correctly predicted as false and 32 as true for high alcohol consumption.
2.3.3.4. Graphic visualization of actual values and predictions
The graph represent the 2x2 cross table in a graphical view with the false negative values on top left of the graph. 82 cases predicted at FALSE but with a TRUE high alcohol use.
| FALSE | TRUE | Sum | |
|---|---|---|---|
| high use | |||
| FALSE | 0.6623 | 0.0393 | 0.7016 |
| TRUE | 0.2147 | 0.0838 | 0.2984 |
| Sum | 0.8770 | 0.1230 | 1.0000 |
Is the prediction of high alcohol use correct of not?
This propability table gives us the correct predictions, false negative and false positive values as fractions of the total number. Results in fraction values between 0 and 1.
So 21.5 % of are false negative predictions! 3.9 % are false positive predictions.
2.3.3.5. Computing the loss function - testing error
To measure the performance of of the logistic regression model we perform the loss function to calculate the average number of wrong predictions of our model.
This results to following value:
## [1] 0.2539267So 25.4% of all predictions are wrong! Every fourth prediction is incorrect (to check that you can calculate the sum of false positive and false negative values from the Target variable versus the predictions table: 21.5 + 3.9 = 25.4).
So on the training data the model has a prediction error of 25.4 %.
2.3.3.6. Cross validation - the training error
Here we are testing the how good our model is on unseen data. The loss function value is computed on a data which we did not use to to train the model. If the value is low it is good! We are performing the cross validation so that we test the logistic regression models on a defined set of observations. Here we choose the value of 10 - so the dataset will be split in 10 groups and 1 group of these 10 will be tested.
# K-fold cross-validation
library(boot)
cross_val<- cv.glm(data = alc, cost = loss_func, glmfit = cv_glm, K = 10)The cross validation results to following value:
# average number of wrong predictions in the cross validation
cross_val$delta[1]## [1] 0.2591623This is the average number of wrong predictions in the cross validation. The prediction error on the testing data is higher compared to the training data!
Be aware of it that this value is changing with each new computing because the function computes everytime another set of observations with the logistic regression model.
The cross validation tells us if the model is too much in the data –> if the model is more generalized the value of cross validation shall be higher than the loss function value.
With the cross validation you check your model with the same data - the outcome should be higher number than the loss function –> that shows that the model is not too much in the data –> so the model is more kind of a general one!
Bonus tasks
Bonus task 1
The 10-fold cross-validation of my model was computed in section 2.3.3.6. above. My logistic model does not have a better performance compared to the DataCamp model (both have an error value of ~ 0.26).
I think a better model could be found - maybe more variables need to be used in the model, for example the age of the students or if students are in a relationship. These factors could make the model more accurate.
Bonus task 2
Cross-validations with different logistic regression models
Model 1: 8 variables (sex, age, Pstatus, guardian, failures, famrel, freetime, absences)
Model 2: 7 variables (sex, age, guardian, failures, famrel, freetime, absences) - Pstatus
Model 3: 6 variables (sex, age, failures, famrel, freetime, absences) - Pstatus, guardian
Model 4: 5 variables (sex, age, failures, freetime, absences) - Pstatus, guardian, famrel
Model 5: 4 variables (sex, age, failures, absences) - Pstatus, guardian, famrel, freetime
Model 6: 3 variables (sex, failures, absences) - Pstatus, guardian, famrel, freetime, age
The graph shows a red line representing the training error values and a green line representing the testing error values. You can see how the error values are changing. As more variables are used for the logistic regression model as higher the training and testing error becomes. It seems the model is then too “deep” in the data.
Chapter 4: Clustering and classification
Data wrangling and performing clustering and classification
Work of week 47 (18.11. - 24.11.2019)
1. Analysis of Boston data set
1.1. Load the data set & check the structure
# load necessary packages
library(MASS) # package includes the Boston data set
library(tidyr)
library(dplyr)
library(corrplot)
library(ggplot2)
library(GGally)
library(knitr)
library(kableExtra)
data(Boston) # load the Boston data set1.2. The data set
The Boston data set consists of 14 variables with 506 observations. The data is bout housing values in Boston suburbs. The data set has following variables (columns):
- crim: the per capita crime rate by town
- zn: proportion of residential land zoned for lots over 25,000 sq.ft
- indus: proportion of non-retail business acres per town
- chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
- nox: nitrogen oxides concentration (parts per 10 million)
- rm: average number of rooms per dwelling
- age: proportion of owner-occupied units built prior to 1940
- dis: weighted mean of distances to five Boston employment centres
- rad: index of accessibility to radial highways
- tax: full-value property-tax rate per $10,000
- ptratio: pupil-teacher ratio by town
- black: 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
- lstat: lower status of the population (percent)
- medv: median value of owner-occupied homes in $1000s
We are interested in the later analysis on the per capita crime rate. Overall the crime rate is low and the data distribution is high on the low crime rate.
str(Boston) # check the structure## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...knitr::kable(head(Boston)) %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") # the data frame head| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.00632 | 18 | 2.31 | 0 | 0.538 | 6.575 | 65.2 | 4.0900 | 1 | 296 | 15.3 | 396.90 | 4.98 | 24.0 |
| 0.02731 | 0 | 7.07 | 0 | 0.469 | 6.421 | 78.9 | 4.9671 | 2 | 242 | 17.8 | 396.90 | 9.14 | 21.6 |
| 0.02729 | 0 | 7.07 | 0 | 0.469 | 7.185 | 61.1 | 4.9671 | 2 | 242 | 17.8 | 392.83 | 4.03 | 34.7 |
| 0.03237 | 0 | 2.18 | 0 | 0.458 | 6.998 | 45.8 | 6.0622 | 3 | 222 | 18.7 | 394.63 | 2.94 | 33.4 |
| 0.06905 | 0 | 2.18 | 0 | 0.458 | 7.147 | 54.2 | 6.0622 | 3 | 222 | 18.7 | 396.90 | 5.33 | 36.2 |
| 0.02985 | 0 | 2.18 | 0 | 0.458 | 6.430 | 58.7 | 6.0622 | 3 | 222 | 18.7 | 394.12 | 5.21 | 28.7 |
knitr::kable(summary(Boston)) %>%
kable_styling(bootstrap_options = "striped", position = "center", font_size = 11) %>%
scroll_box(width = "100%", height = "300px")# summary statistics| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Min. : 0.00632 | Min. : 0.00 | Min. : 0.46 | Min. :0.00000 | Min. :0.3850 | Min. :3.561 | Min. : 2.90 | Min. : 1.130 | Min. : 1.000 | Min. :187.0 | Min. :12.60 | Min. : 0.32 | Min. : 1.73 | Min. : 5.00 | |
| 1st Qu.: 0.08204 | 1st Qu.: 0.00 | 1st Qu.: 5.19 | 1st Qu.:0.00000 | 1st Qu.:0.4490 | 1st Qu.:5.886 | 1st Qu.: 45.02 | 1st Qu.: 2.100 | 1st Qu.: 4.000 | 1st Qu.:279.0 | 1st Qu.:17.40 | 1st Qu.:375.38 | 1st Qu.: 6.95 | 1st Qu.:17.02 | |
| Median : 0.25651 | Median : 0.00 | Median : 9.69 | Median :0.00000 | Median :0.5380 | Median :6.208 | Median : 77.50 | Median : 3.207 | Median : 5.000 | Median :330.0 | Median :19.05 | Median :391.44 | Median :11.36 | Median :21.20 | |
| Mean : 3.61352 | Mean : 11.36 | Mean :11.14 | Mean :0.06917 | Mean :0.5547 | Mean :6.285 | Mean : 68.57 | Mean : 3.795 | Mean : 9.549 | Mean :408.2 | Mean :18.46 | Mean :356.67 | Mean :12.65 | Mean :22.53 | |
| 3rd Qu.: 3.67708 | 3rd Qu.: 12.50 | 3rd Qu.:18.10 | 3rd Qu.:0.00000 | 3rd Qu.:0.6240 | 3rd Qu.:6.623 | 3rd Qu.: 94.08 | 3rd Qu.: 5.188 | 3rd Qu.:24.000 | 3rd Qu.:666.0 | 3rd Qu.:20.20 | 3rd Qu.:396.23 | 3rd Qu.:16.95 | 3rd Qu.:25.00 | |
| Max. :88.97620 | Max. :100.00 | Max. :27.74 | Max. :1.00000 | Max. :0.8710 | Max. :8.780 | Max. :100.00 | Max. :12.127 | Max. :24.000 | Max. :711.0 | Max. :22.00 | Max. :396.90 | Max. :37.97 | Max. :50.00 |
1.2.1 Overview plot and data description
# graphical overview of the Boston data set
ov_boston <- ggpairs(Boston, mapping = aes(), title ="Overview of the Boston data set",
lower = list(combo = wrap("facethist", bins = 20)),
upper = list(continuous = wrap("cor", size = 2.8)))
# overview plot of interesting data
bos_detail <- Boston[,c("crim","dis","tax","medv")]
ov_bos_detail <- ggpairs(bos_detail, mapping = aes(),lower = list(combo = wrap("facethist", bins = 20)),
upper = list(continuous = wrap("cor", size = 5)))
ov_bos_detail
The overview plot of the interesting data (for me) gives following information and the histograms give following information: the crime rate has the highest distribution on the low crime rate level. There might be some relationship with the distance to employment center. The tax rate has a high count on a lower level, decreases with a higher tax and has again a higher count on high taxes. The median value of the owner-occupied homes has a maximum count around 20000$. Tax and crime rate show a correlation of 50%.
1.2.2. Data correlations of the Boston data set
# calculate the correlation matrix and round it
cor_matrix<-cor(Boston) %>% round(digits = 2)
# print the correlation matrix
knitr::kable(cor_matrix, caption="Correlation matrix values") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>%
scroll_box(width = "100%", height = "300px")# summary statistics| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| crim | 1.00 | -0.20 | 0.41 | -0.06 | 0.42 | -0.22 | 0.35 | -0.38 | 0.63 | 0.58 | 0.29 | -0.39 | 0.46 | -0.39 |
| zn | -0.20 | 1.00 | -0.53 | -0.04 | -0.52 | 0.31 | -0.57 | 0.66 | -0.31 | -0.31 | -0.39 | 0.18 | -0.41 | 0.36 |
| indus | 0.41 | -0.53 | 1.00 | 0.06 | 0.76 | -0.39 | 0.64 | -0.71 | 0.60 | 0.72 | 0.38 | -0.36 | 0.60 | -0.48 |
| chas | -0.06 | -0.04 | 0.06 | 1.00 | 0.09 | 0.09 | 0.09 | -0.10 | -0.01 | -0.04 | -0.12 | 0.05 | -0.05 | 0.18 |
| nox | 0.42 | -0.52 | 0.76 | 0.09 | 1.00 | -0.30 | 0.73 | -0.77 | 0.61 | 0.67 | 0.19 | -0.38 | 0.59 | -0.43 |
| rm | -0.22 | 0.31 | -0.39 | 0.09 | -0.30 | 1.00 | -0.24 | 0.21 | -0.21 | -0.29 | -0.36 | 0.13 | -0.61 | 0.70 |
| age | 0.35 | -0.57 | 0.64 | 0.09 | 0.73 | -0.24 | 1.00 | -0.75 | 0.46 | 0.51 | 0.26 | -0.27 | 0.60 | -0.38 |
| dis | -0.38 | 0.66 | -0.71 | -0.10 | -0.77 | 0.21 | -0.75 | 1.00 | -0.49 | -0.53 | -0.23 | 0.29 | -0.50 | 0.25 |
| rad | 0.63 | -0.31 | 0.60 | -0.01 | 0.61 | -0.21 | 0.46 | -0.49 | 1.00 | 0.91 | 0.46 | -0.44 | 0.49 | -0.38 |
| tax | 0.58 | -0.31 | 0.72 | -0.04 | 0.67 | -0.29 | 0.51 | -0.53 | 0.91 | 1.00 | 0.46 | -0.44 | 0.54 | -0.47 |
| ptratio | 0.29 | -0.39 | 0.38 | -0.12 | 0.19 | -0.36 | 0.26 | -0.23 | 0.46 | 0.46 | 1.00 | -0.18 | 0.37 | -0.51 |
| black | -0.39 | 0.18 | -0.36 | 0.05 | -0.38 | 0.13 | -0.27 | 0.29 | -0.44 | -0.44 | -0.18 | 1.00 | -0.37 | 0.33 |
| lstat | 0.46 | -0.41 | 0.60 | -0.05 | 0.59 | -0.61 | 0.60 | -0.50 | 0.49 | 0.54 | 0.37 | -0.37 | 1.00 | -0.74 |
| medv | -0.39 | 0.36 | -0.48 | 0.18 | -0.43 | 0.70 | -0.38 | 0.25 | -0.38 | -0.47 | -0.51 | 0.33 | -0.74 | 1.00 |
# Specialized the insignificant value according to the significant level
p.mat <- cor.mtest(cor_matrix)$p
# visualize the correlation matrix
# correlations / colour shows the correlation values
corrplot(cor_matrix, method="circle", type="upper", tl.cex = 0.6, p.mat = p.mat, sig.level = 0.01, title="Correlations of the Boston data set", mar=c(0,0,1,0)) 
Insignificant values are shown with a cross in the square. The crime rate shows rather strong positive correlation with accessibility to highways, the property-tax rate and with the lower population status.
1.3. Data set analysis
1.3.1. Data standardization (scaling of the data set)
Here we standardize the data set and print out summaries of the scaled data set. The data scaling is a method to standardize the values (some kind of transformation), so that we can compare the observations of the different variables. That creates negative and positive values with an overall mean of 0 and a standard deviation of 1.
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
knitr::kable(summary(boston_scaled), caption="Summary of scaled Boston data set") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center", font_size = 11) %>%
scroll_box(width = "100%", height = "300px")| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Min. :-0.419367 | Min. :-0.48724 | Min. :-1.5563 | Min. :-0.2723 | Min. :-1.4644 | Min. :-3.8764 | Min. :-2.3331 | Min. :-1.2658 | Min. :-0.9819 | Min. :-1.3127 | Min. :-2.7047 | Min. :-3.9033 | Min. :-1.5296 | Min. :-1.9063 | |
| 1st Qu.:-0.410563 | 1st Qu.:-0.48724 | 1st Qu.:-0.8668 | 1st Qu.:-0.2723 | 1st Qu.:-0.9121 | 1st Qu.:-0.5681 | 1st Qu.:-0.8366 | 1st Qu.:-0.8049 | 1st Qu.:-0.6373 | 1st Qu.:-0.7668 | 1st Qu.:-0.4876 | 1st Qu.: 0.2049 | 1st Qu.:-0.7986 | 1st Qu.:-0.5989 | |
| Median :-0.390280 | Median :-0.48724 | Median :-0.2109 | Median :-0.2723 | Median :-0.1441 | Median :-0.1084 | Median : 0.3171 | Median :-0.2790 | Median :-0.5225 | Median :-0.4642 | Median : 0.2746 | Median : 0.3808 | Median :-0.1811 | Median :-0.1449 | |
| Mean : 0.000000 | Mean : 0.00000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | |
| 3rd Qu.: 0.007389 | 3rd Qu.: 0.04872 | 3rd Qu.: 1.0150 | 3rd Qu.:-0.2723 | 3rd Qu.: 0.5981 | 3rd Qu.: 0.4823 | 3rd Qu.: 0.9059 | 3rd Qu.: 0.6617 | 3rd Qu.: 1.6596 | 3rd Qu.: 1.5294 | 3rd Qu.: 0.8058 | 3rd Qu.: 0.4332 | 3rd Qu.: 0.6024 | 3rd Qu.: 0.2683 | |
| Max. : 9.924110 | Max. : 3.80047 | Max. : 2.4202 | Max. : 3.6648 | Max. : 2.7296 | Max. : 3.5515 | Max. : 1.1164 | Max. : 3.9566 | Max. : 1.6596 | Max. : 1.7964 | Max. : 1.6372 | Max. : 0.4406 | Max. : 3.5453 | Max. : 2.9865 |
# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
knitr::kable(boston_scaled, caption="Values of scaled Boston data set") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>% scroll_box(width = "100%", height = "300px")| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| -0.4193669 | 0.2845483 | -1.2866362 | -0.2723291 | -0.1440749 | 0.4132629 | -0.1198948 | 0.1400750 | -0.9818712 | -0.6659492 | -1.4575580 | 0.4406159 | -1.0744990 | 0.1595278 |
| -0.4169267 | -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 0.1940824 | 0.3668034 | 0.5566090 | -0.8670245 | -0.9863534 | -0.3027945 | 0.4406159 | -0.4919525 | -0.1014239 |
| -0.4169290 | -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 1.2814456 | -0.2655490 | 0.5566090 | -0.8670245 | -0.9863534 | -0.3027945 | 0.3960351 | -1.2075324 | 1.3229375 |
| -0.4163384 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.0152978 | -0.8090878 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4157514 | -1.3601708 | 1.1815886 |
| -0.4120741 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.2273620 | -0.5106743 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4406159 | -1.0254866 | 1.4860323 |
| -0.4166314 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 0.2068916 | -0.3508100 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4101651 | -1.0422909 | 0.6705582 |
| -0.4098372 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3880270 | -0.0701592 | 0.8384142 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4263763 | -0.0312367 | 0.0399249 |
| -0.4032966 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.1603069 | 0.9778406 | 1.0236249 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4406159 | 0.9097999 | 0.4965904 |
| -0.3955433 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.9302853 | 1.1163897 | 1.0861216 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3281233 | 2.4193794 | -0.6559463 |
| -0.4003331 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3994130 | 0.6154813 | 1.3283202 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3289995 | 0.6227277 | -0.3949946 |
| -0.3939564 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | 0.1314594 | 0.9138948 | 1.2117800 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3926395 | 1.0918456 | -0.8190411 |
| -0.4064448 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3922967 | 0.5089051 | 1.1547920 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4406159 | 0.0863929 | -0.3949946 |
| -0.4091990 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.5630867 | -1.0506607 | 0.7863653 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3705134 | 0.4280788 | -0.0905509 |
| -0.3468869 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4776917 | -0.2406812 | 0.4333252 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4406159 | -0.6151835 | -0.2318998 |
| -0.3459336 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2684739 | 0.5657458 | 0.3166900 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2557205 | -0.3351131 | -0.4711055 |
| -0.3471625 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6413655 | -0.4289659 | 0.3341188 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4265954 | -0.5857761 | -0.2862647 |
| -0.2975737 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4976172 | -1.3952572 | 0.3341188 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3305330 | -0.8504426 | 0.0616709 |
| -0.3289320 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4193385 | 0.4662746 | 0.2198106 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3294377 | 0.2824421 | -0.5472164 |
| -0.3267801 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.1793541 | -1.1359217 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | -0.7413783 | -0.1348628 | -0.2536457 |
| -0.3357215 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.7936533 | 0.0328645 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3754425 | -0.1922772 | -0.4711055 |
| -0.2745709 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.0171035 | 1.0488914 | 0.0013569 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2179309 | 1.1716657 | -0.9712629 |
| -0.3210451 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4549197 | 0.7327152 | 0.1031753 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3927490 | 0.1648126 | -0.3188837 |
| -0.2768170 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2030044 | 0.8215287 | 0.0863639 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4406159 | 0.8495847 | -0.7972951 |
| -0.3051886 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 1.1163897 | 0.1425445 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4147656 | 1.0120256 | -0.8734060 |
| -0.3328778 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.5132730 | 0.9067897 | 0.2871038 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4124654 | 0.5106995 | -0.7538032 |
| -0.3223820 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.9758293 | 0.6083763 | 0.3132232 | -0.6373311 | -0.6006817 | 1.1753027 | -0.5833190 | 0.5401069 | -0.9386440 |
| -0.3419866 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 0.7717932 | 0.4212153 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2213265 | 0.3020471 | -0.6450733 |
| -0.3089856 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.3382132 | 0.7185050 | 0.3126533 | -0.6373311 | -0.6006817 | 1.1753027 | -0.5508966 | 0.6479340 | -0.8407871 |
| -0.3302353 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | 0.2994029 | 0.9174474 | 0.3132707 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3424724 | 0.0205763 | -0.4493595 |
| -0.3035587 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | 0.5541647 | 0.6652169 | 0.2108350 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2580207 | -0.0942525 | -0.1666618 |
| -0.2886358 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8135788 | 0.9067897 | 0.2079856 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0382932 | 1.3929213 | -1.0691198 |
| -0.2626044 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.3026319 | 1.1163897 | 0.1804414 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2196834 | 0.0541848 | -0.8734060 |
| -0.2587365 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4762685 | 0.4769322 | 0.0925851 | -0.6373311 | -0.6006817 | 1.1753027 | -1.3590472 | 2.1085012 | -1.0147549 |
| -0.2862048 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8306578 | 0.9387626 | -0.0037245 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0229582 | 0.7977717 | -1.0256279 |
| -0.2325982 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2684739 | 1.0062609 | -0.0167367 | -0.6373311 | -0.6006817 | 1.1753027 | -1.1869674 | 1.0764418 | -0.9821359 |
| -0.4126414 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.5004637 | -0.0133185 | -0.2064589 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.4163335 | -0.3949946 |
| -0.4087735 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6314027 | -0.2548913 | -0.1981007 | -0.5224844 | -0.7668172 | 0.3438730 | 0.2287748 | -0.1740726 | -0.2753917 |
| -0.4107848 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6185935 | -0.9618471 | 0.0660857 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.5437656 | -0.1666618 |
| -0.3997507 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.4534965 | -1.3632843 | 0.0248170 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4026072 | -0.3533177 | 0.2356387 |
| -0.4168895 | 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 0.4417279 | -1.6616978 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4267049 | -1.1669222 | 0.8988910 |
| -0.4161966 | 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 1.0523023 | -1.8748503 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4265954 | -1.4946046 | 1.3446835 |
| -0.4052857 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | 0.6907967 | -2.3331282 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3147600 | -1.0941039 | 0.4422255 |
| -0.4036511 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1645767 | -2.2016841 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.2924148 | -0.9582698 | 0.3008766 |
| -0.4015748 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1048002 | -2.2052367 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4138893 | -0.7300124 | 0.2356387 |
| -0.4058380 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.3069017 | -1.0151352 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3583550 | -0.4345381 | -0.1449159 |
| -0.4001727 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.8576995 | -1.2353928 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | -0.3421149 | -0.3515026 |
| -0.3982033 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.7096815 | -1.2531555 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.2096238 | -0.2753917 |
| -0.3934472 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.3624084 | 0.6012712 | 0.8996287 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3950493 | 0.8607875 | -0.6450733 |
| -0.3905872 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -1.2604793 | 0.9494202 | 0.9853955 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 2.5426103 | -0.8842790 |
| -0.3945516 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.9715595 | -0.2335761 | 1.0887811 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.4966960 | -0.3406296 |
| -0.4097861 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4577662 | -0.8126404 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4259382 | 0.1115992 | -0.3080107 |
| -0.4150596 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.2414322 | -0.1980507 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4085221 | -0.4513423 | -0.2210268 |
| -0.4138702 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | 0.3221749 | -1.6865656 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -1.0324884 | 0.2682577 |
| -0.4143109 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4079525 | -1.6759080 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -0.5913775 | 0.0942899 |
| -0.4185206 | 2.7285450 | -1.0402932 | -0.2723291 | -1.2486880 | -0.5645100 | -0.7451421 | 1.6738568 | -0.7521778 | 0.3605309 | 1.2214933 | 0.4406159 | 0.3006467 | -0.3949946 |
| -0.4185775 | 3.3717021 | -1.4455202 | -0.2723291 | -1.3090965 | 1.3725336 | -1.6581453 | 2.3277455 | -0.5224844 | -1.0812880 | -0.2566040 | 0.4299910 | -1.0983050 | 1.3990484 |
| -0.4177126 | 3.1573164 | -1.5154874 | -0.2723291 | -1.2486880 | 0.1399989 | -1.1678945 | 2.5609210 | -0.8670245 | -0.5650812 | -0.5337472 | 0.4406159 | -0.9638712 | 0.2356387 |
| -0.4184369 | 3.8004735 | -1.4309437 | -0.2723291 | -1.2400582 | 0.7562662 | -0.9973725 | 2.1511780 | -0.5224844 | -0.9032856 | -1.5499390 | 0.3968018 | -1.2187352 | 0.9858749 |
| -0.4021456 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.1987347 | -1.3988097 | 1.9089794 | -0.1779443 | -0.7371501 | 0.5748257 | 0.3724850 | -0.8112328 | 0.0834169 |
| -0.4080945 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.5090032 | -0.7593522 | 1.4897384 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.4807497 | -0.3188837 |
| -0.4027420 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.7737278 | -0.0843694 | 1.6290739 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4210091 | 0.0695886 | -0.4167406 |
| -0.4001390 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.4534965 | 0.8819220 | 1.4358374 | -0.1779443 | -0.7371501 | 0.5748257 | 0.2344707 | 0.2502341 | -0.7103112 |
| -0.4072819 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.2438961 | -0.0275287 | 1.6291213 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.8294374 | -0.0361860 |
| -0.4053950 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.6794107 | -0.8943488 | 1.9878602 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4261573 | -0.4415399 | 0.2682577 |
| -0.4178335 | 0.2631097 | -1.4221978 | -0.2723291 | -1.1960462 | 1.1661623 | -0.3223896 | 2.5776850 | -0.7521778 | -1.1406221 | 0.0667298 | 0.4005260 | -0.6445909 | 1.1380967 |
| -0.4159350 | 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | 0.0076366 | -1.8037995 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -1.1179099 | 0.1051628 |
| -0.4150107 | 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | -0.7082582 | -1.3313114 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -0.3379138 | -0.3406296 |
| -0.4133715 | 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5787425 | -1.6759080 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4330580 | -0.6375891 | -0.0579320 |
| -0.4043440 | 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.9829455 | -1.1288166 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | 0.0611865 | -0.5580894 |
| -0.4052020 | 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5687797 | -1.2638131 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | -0.5409648 | -0.1775348 |
| -0.4098407 | -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | 0.1883894 | -2.2016841 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2963581 | -0.8308377 | 0.1812738 |
| -0.4016445 | -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.4606127 | -1.8144571 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2219837 | -0.3883265 | -0.0905509 |
| -0.4094478 | -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.3125947 | -2.1590536 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.3750043 | -0.9988800 | 0.0290519 |
| -0.3973860 | -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.0564097 | -2.2158943 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2245030 | -0.7160089 | 0.0942899 |
| -0.4109219 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0165586 | -2.2229994 | 0.2167712 | -0.5224844 | -0.0607412 | 0.1129203 | 0.4189279 | -0.8224356 | 0.1704008 |
| -0.4090432 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | 0.0019436 | -0.8375082 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.2908813 | -0.5199596 | -0.1231699 |
| -0.4082980 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0080191 | 0.2104916 | 0.1221238 | -0.5224844 | -0.0607412 | 0.1129203 | 0.1860561 | -0.0956529 | -0.2753917 |
| -0.4099791 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.2058509 | -0.8090878 | 0.1403124 | -0.5224844 | -0.0607412 | 0.1129203 | 0.3317379 | -0.3337127 | -0.1884078 |
| -0.4135377 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0749119 | -0.5284370 | 0.5789293 | -0.5224844 | -0.0607412 | 0.1129203 | 0.3256039 | -0.0438399 | -0.1449159 |
| -0.4103511 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.5844355 | -1.1359217 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.4314149 | -0.4975539 | -0.2427728 |
| -0.4153200 | 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.6295970 | -1.2460504 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4406159 | -1.0310881 | 0.5944473 |
| -0.4149142 | 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.4758859 | 0.0648374 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4267049 | -0.7608201 | 0.1486548 |
| -0.4158478 | 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.0247156 | -1.2922335 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4406159 | -0.8308377 | 0.2465117 |
| -0.4159734 | 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | -0.1674232 | -0.7771150 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.3720469 | -0.7202099 | 0.0399249 |
| -0.4142202 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.1485384 | -0.7309319 | 0.4674705 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4406159 | -0.4247356 | 0.1486548 |
| -0.4134343 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.4915417 | -0.4431760 | 0.3051974 | -0.7521778 | -0.9566863 | 0.0205393 | 0.3902297 | -0.8574444 | 0.4422255 |
| -0.4140702 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | -0.3837572 | -0.8339556 | 0.3002110 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4306482 | 0.0289784 | -0.0035670 |
| -0.4117881 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | -0.2328927 | -0.4183083 | -0.0225305 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4214472 | -0.5899772 | -0.0361860 |
| -0.4135215 | -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 1.0281070 | 0.6296915 | -0.1773001 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4406159 | -1.0016807 | 0.1160358 |
| -0.4139377 | -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 1.1305810 | -0.1944981 | -0.1807194 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4314149 | -0.9736736 | 0.6705582 |
| -0.4146561 | -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 0.1883894 | -0.0879219 | -0.3337319 | -0.8670245 | -0.8202179 | -0.3027945 | 0.3889153 | -0.5381641 | 0.0073060 |
| -0.4155304 | -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 0.1713104 | 0.1891763 | -0.3338269 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4039216 | -0.6235856 | -0.0579320 |
| -0.4152153 | 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | 0.2239706 | -0.5319896 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4199137 | -0.6291870 | 0.0399249 |
| -0.4167593 | 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.1048002 | -1.4094674 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4343724 | -0.9022557 | 0.2682577 |
| -0.4151096 | 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.0507166 | 0.3099628 | -0.0855021 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4406159 | -0.2889015 | -0.2101538 |
| -0.4059135 | -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 0.4844254 | -0.3827828 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.0143049 | -0.8406402 | 0.6379392 |
| -0.4067273 | -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | -0.1731162 | 0.0364171 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.3850816 | -0.1838751 | -0.1231699 |
| -0.4060542 | -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 2.5395987 | 0.2637797 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4406159 | -1.1823261 | 1.7578570 |
| -0.4105836 | -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 2.1852094 | -1.1252640 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4037025 | -1.2719486 | 2.3123794 |
| -0.4121264 | -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 1.6102164 | -0.2158134 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4406159 | -0.9050564 | 1.1598427 |
| -0.4028187 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.6295970 | 0.4023288 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4171754 | -0.4527427 | 0.5400824 |
| -0.4068110 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.7064525 | 0.0968103 | -0.4459031 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4261573 | -0.6978043 | 0.4313525 |
| -0.3935065 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.1713104 | 0.5977186 | -0.5130539 | -0.5224844 | -0.1438090 | 1.1291122 | -3.1313265 | -0.2833001 | -0.4276135 |
| -0.3955003 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.2101207 | 0.6687695 | -0.5130539 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4139988 | 0.1101988 | -0.3515026 |
| -0.4038720 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1674232 | 0.7611355 | -0.6525317 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3945016 | -0.0452402 | -0.2645187 |
| -0.4046835 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6171702 | 0.9991558 | -0.8016976 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4093984 | 0.5345055 | -0.3297567 |
| -0.4001983 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6385190 | 0.8286338 | -0.7522606 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4271431 | 0.8411826 | -0.3297567 |
| -0.4048521 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.2243532 | 0.5906135 | -0.7943366 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3397340 | 0.2012217 | -0.2318998 |
| -0.4052183 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.2695146 | 1.0133660 | -0.6468804 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4224331 | -0.0536423 | -0.2971377 |
| -0.3894525 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.0791817 | 0.8037660 | -0.5935968 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3785094 | 0.4056731 | -0.3406296 |
| -0.4075539 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1275722 | -0.5035693 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4032644 | 0.0485834 | -0.0905509 |
| -0.4083782 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | 0.6125180 | 0.4627220 | -0.5307201 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4262668 | -0.3491166 | 0.0290519 |
| -0.4057682 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5289287 | 0.8641592 | -0.6846349 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4192565 | 0.4980964 | -0.4058676 |
| -0.3942784 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.2741669 | 0.9529728 | -0.5922195 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4406159 | 0.6213273 | -0.4167406 |
| -0.4035570 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.0436004 | 0.5550881 | -0.7306527 | -0.4076377 | 0.1409947 | -0.3027945 | 0.3512352 | -0.3085064 | -0.4384865 |
| -0.4001820 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5075800 | 0.6971898 | -0.6325385 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1288575 | 0.4350805 | -0.4602325 |
| -0.4048044 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.1546139 | 0.1394408 | -0.5057404 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4011832 | -0.0858504 | -0.1449159 |
| -0.4025490 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.3752177 | 0.4982475 | -0.4975246 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4144370 | -0.3295117 | -0.3623756 |
| -0.4049207 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5872820 | 0.1607560 | -0.6256999 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1976456 | 0.3804668 | -0.2318998 |
| -0.4032721 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.7879603 | -0.1198948 | -0.4919208 | -0.4076377 | 0.1409947 | -0.3027945 | 0.3814669 | 0.1340048 | -0.3515026 |
| -0.4120810 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.5901285 | 0.0399696 | -0.7300828 | -0.8670245 | -1.3067576 | 0.2976825 | 0.3557261 | 0.2404316 | -0.0579320 |
| -0.4117718 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.3994130 | 0.5515356 | -0.7587192 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2299797 | 0.2264281 | -0.2427728 |
| -0.4092908 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.4606127 | 0.8641592 | -0.8111956 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2345802 | 0.7389569 | -0.2210268 |
| -0.4026188 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.6100540 | 1.0098134 | -0.8788687 | -0.8670245 | -1.3067576 | 0.2976825 | 0.1493618 | 1.7864202 | -0.5689624 |
| -0.4086514 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.5773192 | 0.9671829 | -0.8494724 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2487102 | 0.6899446 | -0.4058676 |
| -0.4004517 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.4250315 | 0.7042949 | -0.8558361 | -0.8670245 | -1.3067576 | 0.2976825 | 0.3104881 | 0.3020471 | -0.1231699 |
| -0.3750691 | -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.9559038 | 0.9600779 | -0.9677698 | -0.8670245 | -1.3067576 | 0.2976825 | 0.0286541 | 2.0454854 | -0.7429302 |
| -0.3899734 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.8420438 | 0.9742880 | -0.9530004 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3881485 | 0.6353309 | -0.6885653 |
| -0.3822678 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2083149 | 1.0737592 | -0.9415079 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.3832675 | -0.4928515 |
| -0.3176492 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.9217458 | 0.9281050 | -0.8620098 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.7963713 | -0.8951520 |
| -0.3805669 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2467426 | 1.0773117 | -0.7961887 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4202424 | -0.0074307 | -0.3623756 |
| -0.2814126 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.0588736 | 1.0346812 | -0.7237666 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | -0.0550427 | -0.3188837 |
| -0.3515035 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.1243431 | 1.0417863 | -0.6969823 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3185937 | -0.2146828 | 0.0507979 |
| -0.3817574 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.6584445 | 0.9529728 | -0.6293092 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3506875 | 0.3328548 | -0.4493595 |
| -0.3066139 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.7509558 | 1.0595490 | -0.6881492 | -0.6373311 | 0.1706618 | 1.2676838 | -1.0286891 | 0.6521351 | -0.7538032 |
| -0.3552552 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.0716829 | 1.0524439 | -0.7998930 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4161895 | 0.6031228 | -0.4819785 |
| -0.3825921 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.4876545 | 0.8854745 | -0.8681835 | -0.6373311 | 0.1706618 | 1.2676838 | 0.2363328 | 0.5947207 | -0.5580894 |
| -0.3791404 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2410496 | 1.0595490 | -0.9237941 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4097270 | 0.2712393 | -0.5907084 |
| -0.3910604 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.6086307 | 1.0524439 | -1.0098459 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3873818 | 1.2136763 | -1.0038819 |
| -0.3567968 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.1901952 | 1.0417863 | -1.0097984 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.8131756 | -0.5145975 |
| -0.3862822 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.1574604 | 0.8890270 | -1.0367727 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3440059 | 1.6113762 | -0.9277710 |
| -0.2307590 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -1.8013144 | 1.1163897 | -1.1186928 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 3.0467371 | -0.8842790 |
| -0.0340024 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -1.2547863 | 1.1163897 | -1.1746359 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 1.9838699 | -0.9930089 |
| 0.0562546 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.1622751 | 1.1163897 | -1.1318000 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 1.9278558 | -0.7538032 |
| -0.0969342 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.9664114 | 1.0382338 | -1.1630958 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 2.3297568 | -1.1669767 |
| -0.1434839 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.2200834 | 1.1163897 | -1.1283332 | -0.5224844 | -0.0310742 | -1.7347012 | -2.0128627 | 2.1211044 | -0.9495170 |
| -0.1695595 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.9345550 | 1.1163897 | -1.0820306 | -0.5224844 | -0.0310742 | -1.7347012 | -2.0527336 | 0.5597119 | -0.7538032 |
| -0.1447302 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.9336767 | 0.9636304 | -1.1085299 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3837671 | 2.3633653 | -0.8625331 |
| -0.1491050 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.5636316 | 0.8961321 | -1.0758569 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0034610 | 2.1939227 | -0.5145975 |
| -0.1022553 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.9786758 | 0.9352101 | -1.0777090 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0528401 | 1.2318808 | -0.7755492 |
| -0.2275084 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.2314694 | 1.0204711 | -1.0338758 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1766361 | 0.2026221 | -0.1122969 |
| -0.2461422 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.2533631 | 1.1163897 | -1.0464131 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1651137 | 0.0877932 | -0.3188837 |
| -0.2891275 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -1.8112772 | 0.6900847 | -1.0375800 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1467118 | -0.0746476 | -0.7864221 |
| -0.1702419 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.8192718 | 1.0631016 | -1.0314063 | -0.5224844 | -0.0310742 | -1.7347012 | -1.0375614 | 0.4392816 | -0.3406296 |
| -0.2557300 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.2215067 | 0.9742880 | -0.9714740 | -0.5224844 | -0.0310742 | -1.7347012 | -0.3905371 | 0.3454580 | -0.6015814 |
| -0.0091278 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.1887719 | 0.4982475 | -0.9733261 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9428165 | 0.3314545 | -0.7538032 |
| -0.1356551 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.4412321 | 0.9032372 | -0.9776477 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9360253 | 0.4882939 | -1.0256279 |
| -0.2778505 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.9370190 | 1.0240236 | -0.9107344 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0740016 | -1.1291127 | 2.0405547 |
| -0.2639855 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.3111714 | 1.1163897 | -0.9677223 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0304949 | -0.8714479 | 0.1921468 |
| -0.2544314 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | 0.3207517 | 1.1163897 | -0.9636382 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0836407 | -0.7370141 | 0.0834169 |
| -0.2720515 | -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | -0.0492934 | 0.8535016 | -0.9482040 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1944691 | -1.0016807 | 0.4857174 |
| -0.2499741 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 1.7141136 | 0.7895559 | -0.8662839 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1944903 | -1.5296134 | 2.9865046 |
| -0.2069109 | -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | 2.1595909 | 1.0524439 | -0.8331359 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3607647 | -1.5030067 | 2.9865046 |
| -0.2435032 | -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | 2.9751133 | 0.8996847 | -0.7755306 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3480587 | -1.3069574 | 2.9865046 |
| -0.1594090 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.6129005 | 0.8250813 | -0.6520568 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4210091 | -0.1418645 | 0.0181789 |
| -0.0801628 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.2613577 | 0.8677118 | -0.7178779 | -0.5224844 | -0.0310742 | -1.7347012 | -1.2762386 | -0.3981289 | 0.2682577 |
| -0.1864006 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 2.3403437 | 0.9813931 | -0.8306664 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1382988 | -1.2537440 | 2.9865046 |
| -0.2108044 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5801657 | 0.3774611 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -1.4137053 | -0.0718469 | 0.1377818 |
| -0.1526614 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.0489109 | 0.9778406 | -0.8049744 | -0.5224844 | -0.0310742 | -1.7347012 | -0.6526548 | -0.2174835 | 0.1377818 |
| -0.1353238 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.1670406 | 0.9458677 | -0.7278033 | -0.5224844 | -0.0310742 | -1.7347012 | -0.2917364 | -0.1866758 | -0.0253130 |
| -0.2797292 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5830122 | 0.9245525 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -0.7052317 | 0.2488337 | -0.5580894 |
| -0.1510919 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5758960 | 1.0204711 | -0.6678710 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0935872 | -0.0872508 | -0.3732486 |
| -0.4039255 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -1.0142570 | 0.7078474 | -0.5693769 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4406159 | 0.2852429 | 0.0616709 |
| -0.4094315 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.1869661 | 0.5515356 | -0.5455370 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4252810 | -0.5059560 | 0.1160358 |
| -0.4102814 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -0.6057842 | 0.0044442 | -0.5191326 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4004165 | -0.4219349 | 0.0073060 |
| -0.4123542 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.3719887 | -1.2602606 | -0.3147360 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3755520 | -1.0254866 | 0.7466691 |
| -0.4119380 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -0.3766409 | -0.7593522 | -0.1140436 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4004165 | -0.3561184 | 0.0725439 |
| -0.4137947 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.0432179 | 0.1714136 | -0.2267846 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4263763 | -0.8910529 | 0.2247657 |
| -0.4123798 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.8188892 | 0.2069391 | -0.4177891 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3789476 | -0.8028307 | 0.8010341 |
| -0.4133820 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.9896793 | -0.3614676 | -0.4587729 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -1.0660969 | 1.5947622 |
| -0.4124426 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 2.1069307 | 0.5231153 | -0.5005640 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4259382 | -0.7132081 | 1.8774599 |
| -0.4120938 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.2001579 | -0.2264710 | -0.5685221 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.4485416 | 1.4860323 |
| -0.4095187 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 1.2387480 | 0.8392915 | -0.5197499 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4101651 | -1.0969046 | 1.6708731 |
| -0.4084666 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.3961839 | 0.9600779 | -0.4502247 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.9764743 | 1.0837317 |
| -0.4104430 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.9687130 | 0.7540305 | -0.3833114 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3759901 | 0.1858179 | 0.4204795 |
| -0.4130715 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.1873487 | 0.0079967 | -0.2447358 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3333809 | 0.0695886 | 0.7684151 |
| -0.4135889 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 2.2008652 | -0.5319896 | -0.2829652 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3938444 | -1.1487176 | 2.9865046 |
| -0.4109463 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.7078757 | -0.9760573 | -0.0030596 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4074267 | -0.8364391 | 1.0293668 |
| -0.4054776 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.3862212 | -1.4023623 | 0.3664594 | -0.5224844 | -0.0607412 | -1.5037485 | 0.2866094 | -1.1333138 | 0.7901611 |
| -0.4103709 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 1.2814456 | -1.0542132 | 0.3664594 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4406159 | -1.0170845 | 1.3446835 |
| -0.4095594 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.9484050 | -1.6723554 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.2300893 | -1.0576947 | 1.5730162 |
| -0.4120671 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.6466760 | -1.3419691 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3618601 | -1.1151092 | 0.8662720 |
| -0.4100291 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 1.2714828 | -1.5018334 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3704038 | -1.3699732 | 1.5077783 |
| -0.4175591 | 2.0853880 | -1.1962619 | -0.2723291 | -1.3263561 | 0.7334942 | -2.0844502 | 1.1514203 | -0.9818712 | -0.8498849 | -1.3189864 | 0.4019500 | -1.0674972 | 0.9315099 |
| -0.4184287 | 2.0853880 | -1.1962619 | -0.2723291 | -1.3263561 | 0.4545372 | -1.7682741 | 1.1514203 | -0.9818712 | -0.8498849 | -1.3189864 | 0.2193548 | -1.1585201 | 0.7140502 |
| -0.4184962 | 2.9429307 | -1.5563017 | -0.2723291 | -1.1451305 | 2.2634882 | -1.2993386 | 0.8801579 | -0.6373311 | -0.9092190 | -1.8732728 | 0.4113700 | -1.3559697 | 2.9865046 |
| -0.4154386 | 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.4266171 | -1.2247352 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | 0.4406159 | -1.2005307 | 1.1707156 |
| -0.4146771 | 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.1704320 | -1.1359217 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | -0.0258945 | -0.5661712 | 0.8445260 |
| -0.4157211 | 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.4081148 | -1.0755284 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | 0.3891344 | -0.8448412 | 1.3120645 |
| -0.4164395 | 3.5860878 | -1.4090789 | -0.2723291 | -1.3090965 | 0.9825630 | -1.8926130 | 1.8323307 | -0.7521778 | -0.0370076 | -0.6723188 | 0.4406159 | -1.1333138 | 1.3446835 |
| -0.4180346 | 3.5860878 | -1.4090789 | -0.2723291 | -1.3090965 | 1.2102830 | -1.9423486 | 1.8323307 | -0.7521778 | -0.0370076 | -0.6723188 | 0.3026016 | -1.1487176 | 1.1272237 |
| -0.4160966 | 3.0501236 | -1.3274505 | -0.2723291 | -1.2055390 | -0.1745394 | -1.0719759 | 1.1753552 | -0.8670245 | -0.3574118 | -1.7347012 | 0.4063314 | -0.7314127 | 0.1704008 |
| -0.4175707 | 3.0501236 | -1.3274505 | -0.2723291 | -1.2055390 | 1.8863269 | -1.8784028 | 1.1753552 | -0.8670245 | -0.3574118 | -1.7347012 | 0.4239665 | -1.3363648 | 2.1492845 |
| -0.4160210 | 3.5860878 | -1.2327031 | -0.2723291 | -1.1960462 | 2.2321767 | -1.2567081 | 0.6282713 | -0.6373311 | -1.0931548 | -1.7347012 | 0.3954874 | -1.2383402 | 2.8234098 |
| -0.4177661 | 3.5860878 | -1.2327031 | -0.2723291 | -1.1960462 | 2.4897850 | -1.3028911 | 0.6282713 | -0.6373311 | -1.0931548 | -1.7347012 | 0.3710611 | -1.3685729 | 2.9865046 |
| -0.4042417 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.5602402 | -1.6439351 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | -0.2496916 | 0.0073060 |
| -0.3933983 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.0588736 | -0.5710675 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4183803 | -0.2356881 | 0.2030197 |
| -0.3908058 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.7139512 | 0.1465458 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3587931 | 0.7571615 | -0.0035670 |
| -0.4043057 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.3140179 | -0.3365998 | 0.2109299 | -0.6373311 | -0.7786840 | 0.0667298 | 0.2699601 | 0.2810418 | 0.2030197 |
| -0.3694468 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.3387581 | 1.1163897 | 0.0379717 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | 1.4615386 | -0.2753917 |
| -0.3998193 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.4620360 | 0.8357389 | 0.0389689 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4006356 | 0.6465337 | -0.0905509 |
| -0.3764142 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.2533631 | 0.7114000 | -0.0617572 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4224331 | 1.5861699 | -0.3515026 |
| -0.3948516 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.6797932 | -0.5248845 | -0.0676459 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3753329 | 0.4728900 | -0.0144400 |
| -0.4037651 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.1286129 | -1.2886809 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3191414 | -0.4583441 | 0.6053203 |
| -0.3864391 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -1.2419771 | -2.0880028 | -0.0985619 | -0.6373311 | -0.7786840 | 0.0667298 | -0.0848244 | 2.3661660 | 0.1269088 |
| -0.3970802 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.1460744 | -0.9298742 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4047979 | -0.4457409 | 0.2682577 |
| -0.4148003 | -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | -0.5645100 | -0.4467286 | -0.3243289 | -0.5224844 | -0.7846174 | -0.9494620 | 0.3957065 | 0.1200013 | 0.0834169 |
| -0.4119485 | -0.4872402 | 0.4013236 | -0.2723291 | -0.0405174 | 0.5086207 | 0.5870610 | -0.1775851 | -0.5224844 | -0.7846174 | -0.9494620 | 0.3954874 | -0.4149332 | 0.6705582 |
| -0.4072331 | -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | -0.4748452 | 0.8961321 | -0.4301365 | -0.5224844 | -0.7846174 | -0.9494620 | 0.4406159 | 0.7375566 | -0.1122969 |
| -0.4068192 | -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | 0.1257664 | 0.8463965 | -0.2050342 | -0.5224844 | -0.7846174 | -0.9494620 | 0.4060028 | -0.3015046 | 0.0507979 |
| -0.3784708 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.9484050 | 0.7078474 | -0.4432437 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3836576 | -0.4121325 | 0.4530985 |
| -0.3727021 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | -0.1716929 | 0.8073186 | -0.3547700 | -0.1779443 | -0.6006817 | -0.4875567 | 0.4224331 | 1.2332812 | -0.0905509 |
| -0.3476077 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.8459310 | 0.3241729 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3693085 | -0.3813247 | 0.5400824 |
| -0.3486378 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 0.4744627 | 0.4343017 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.4406159 | -0.7076067 | 0.8227800 |
| -0.3834420 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 2.8199790 | 0.3454882 | -0.4277145 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3108167 | -1.1921285 | 2.4211092 |
| -0.3588418 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 3.4732509 | 0.5124576 | -0.4277145 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2774085 | -1.1235113 | 2.9865046 |
| -0.3756748 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 2.4983245 | 0.6367966 | -0.2751294 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3363384 | -1.3335641 | 1.6382541 |
| -0.3721591 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.2501340 | 0.4023288 | -0.2751294 | -0.1779443 | -0.6006817 | -0.4875567 | 0.1687496 | -0.8812504 | 0.9858749 |
| -0.3854347 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.9944939 | -1.8322198 | -0.1994304 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2282272 | -1.2229363 | 2.6276960 |
| -0.3687411 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 0.3805282 | -1.6759080 | -0.1994304 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2592256 | -1.2453419 | 0.9750019 |
| -0.3576710 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | -0.4321477 | -0.0168711 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2374281 | -0.1404642 | 0.1921468 |
| -0.3662788 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.6045233 | 0.2957526 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2132208 | -1.0366895 | 0.9967478 |
| -0.3532195 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.9210298 | 0.1678611 | 0.0205904 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3202367 | -1.4259873 | 2.0840466 |
| -0.3815656 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.7929373 | 0.0648374 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2440002 | -1.2187352 | 2.8016638 |
| -0.3680285 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.6281737 | -0.0737117 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.0386218 | -0.6445909 | 0.7031772 |
| -0.3816842 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | -0.2827064 | -0.2513388 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2199025 | -0.2482913 | 0.1595278 |
| -0.3595800 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.4929649 | 0.2815424 | 0.1676191 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3480587 | -0.4359384 | 0.2791307 |
| -0.3605973 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 1.5276678 | 0.1074679 | 0.1676191 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3658034 | -1.1095078 | 0.9750019 |
| -0.4105174 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.2794774 | -1.7789317 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2490389 | -0.8812504 | 0.1269088 |
| -0.4093455 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.4573837 | -0.9369793 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2969057 | -0.7398148 | 0.0834169 |
| -0.4069308 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.8715495 | -0.5071218 | 1.2067460 | -0.4076377 | -0.6422155 | -0.8570810 | 0.3787285 | -0.1782737 | -0.0579320 |
| -0.4077644 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | -0.2698972 | -0.1234473 | 1.2067460 | -0.4076377 | -0.6422155 | -0.8570810 | 0.4156419 | -0.0354378 | -0.2645187 |
| -0.4081387 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.1044176 | -0.5568574 | 1.5388905 | -0.4076377 | -0.6422155 | -0.8570810 | 0.1760884 | -0.2006793 | -0.0361860 |
| -0.4052706 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.1542314 | -2.1590536 | 1.5388905 | -0.4076377 | -0.6422155 | -0.8570810 | 0.1975573 | -1.0450916 | 0.1269088 |
| -0.3961432 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9843688 | 0.2815424 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1732405 | -0.0214342 | -0.5363434 |
| -0.3978580 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9672898 | 0.0577323 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3555071 | 0.8131756 | -0.4384865 |
| -0.3805937 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.2513949 | -1.1963149 | 2.0232877 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3670082 | -0.4891518 | 0.1921468 |
| -0.3972488 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.0834514 | 0.3774611 | 2.0232877 | -0.2927910 | -0.4642132 | 0.2976825 | 0.2132208 | -0.3505170 | -0.2210268 |
| -0.4009900 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2111614 | -0.6918540 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1975573 | -0.4387391 | 0.2138927 |
| -0.3979278 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.6167877 | -1.8144571 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4060028 | -0.8532433 | 0.3987335 |
| -0.4037907 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2880169 | -1.9743215 | 1.7104241 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4338247 | -0.9456666 | 0.2030197 |
| -0.3952120 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2182776 | -2.1199757 | 1.7104241 | -0.2927910 | -0.4642132 | 0.2976825 | 0.2234076 | -1.2691479 | 0.2465117 |
| -0.4105441 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.9569445 | -2.1945790 | 2.4275218 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3222084 | -1.2775500 | 0.7684151 |
| -0.3772094 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 2.8100163 | -2.1377384 | 2.4275218 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4406159 | -1.2761497 | 2.2036495 |
| -0.4144992 | 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.2513949 | -1.2993386 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.3966923 | -0.8518430 | -0.0688050 |
| -0.4159768 | 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.5815890 | -1.7576164 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.4217758 | -0.4765487 | -0.1775348 |
| -0.4183136 | 3.3717021 | -1.0767345 | -0.2723291 | -1.3867646 | 1.6642999 | -1.2211827 | 1.2067460 | -0.7521778 | -0.9744866 | -1.1804147 | 0.3249467 | -1.3363648 | 2.3341253 |
| -0.3490052 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 3.4433626 | 0.6510068 | -0.9469692 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3617506 | -1.0548940 | 2.9865046 |
| -0.3429632 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.4920866 | 1.1163897 | -0.9025187 | -0.5224844 | -0.8558183 | -2.5199404 | 0.2915385 | -0.6810000 | 1.4642863 |
| -0.3437607 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 0.7932707 | 1.1163897 | -0.8473829 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3861769 | -0.8056314 | 0.8227800 |
| -0.3573095 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.3070641 | 0.4698271 | -0.7992281 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3957065 | -0.4289367 | 1.2250806 |
| -0.3580059 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.7582344 | 0.7398203 | -0.7860734 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3471824 | -0.7552187 | 2.2362684 |
| -0.3596311 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 3.0078481 | 0.8144237 | -0.7154559 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3306426 | -0.9442662 | 2.8560287 |
| -0.3241585 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.4835471 | 0.9209999 | -0.8150422 | -0.5224844 | -0.8558183 | -2.5199404 | 0.4024977 | -0.1964782 | 0.9206369 |
| -0.3561515 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.3113338 | 0.8179762 | -0.8856597 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3419247 | -0.6375891 | 1.5186513 |
| -0.3315571 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | -1.0313360 | -0.2051558 | -0.8588754 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3913251 | -0.3085064 | 0.0290519 |
| -0.3287576 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.0380698 | 0.5692983 | -0.7893502 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3000823 | 0.2992464 | 0.8880180 |
| -0.3528649 | 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 2.8640998 | -0.0559490 | -0.6522468 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3052304 | -0.7300124 | 2.9865046 |
| -0.3572641 | 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 1.6870719 | -0.5675150 | -0.4383522 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3683227 | -1.3293630 | 2.2797604 |
| -0.4095629 | 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | -0.5189660 | -0.2513388 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3797143 | 0.1396062 | -0.1992808 |
| -0.3853219 | 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.6100540 | -0.9405319 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3502494 | 0.0485834 | -0.1557889 |
| -0.4012551 | 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.0635259 | -1.8570876 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.4406159 | -0.8490423 | 0.2900036 |
| -0.4067785 | 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | 0.3606027 | -0.3508100 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.4193661 | -0.6894022 | 0.2030197 |
| -0.3943063 | 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | 2.0016102 | -0.5959353 | 0.2713846 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3734708 | -0.8504426 | 1.3773024 |
| -0.4135401 | 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 0.6737177 | -1.2673657 | 0.1341862 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -1.2775500 | 1.0728588 |
| -0.4089362 | 1.2278453 | -0.6889993 | -0.2723291 | -0.9293857 | 0.8103497 | -0.9156641 | 0.2242746 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -1.3545694 | 1.0293668 |
| -0.4079306 | 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 1.3981521 | -0.6954065 | 0.4711747 | -0.6373311 | -0.9151524 | -0.3951756 | 0.3568215 | -0.9246613 | 1.1598427 |
| -0.4129785 | 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 0.7704987 | -1.4556504 | 0.5070771 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4028263 | -1.1893278 | 1.1489697 |
| -0.4108266 | 1.2278453 | -0.6889993 | -0.2723291 | -0.9293857 | 0.2809007 | -1.2957860 | 0.1639624 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -0.7650212 | 0.7140502 |
| -0.3956433 | 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 0.7505732 | -1.2922335 | 0.1451564 | -0.5224844 | -1.1406221 | -1.6423201 | 0.4406159 | -1.0927035 | 1.3664294 |
| -0.4159420 | 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 2.1852094 | -0.1447626 | 0.4272465 | -0.5224844 | -1.1406221 | -1.6423201 | 0.3355717 | -1.2453419 | 2.4863472 |
| -0.4157943 | 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 0.9726003 | -1.1146064 | 0.6884411 | -0.5224844 | -1.1406221 | -1.6423201 | 0.3894630 | -1.1291127 | 1.3990484 |
| -0.4129762 | 0.3703025 | -1.1379558 | 3.6647712 | -0.9647679 | 1.9361406 | -0.6705387 | 0.6728644 | -0.5224844 | -1.1406221 | -1.6423201 | 0.2234076 | -1.3503683 | 2.5515851 |
| -0.4183566 | 3.3717021 | -1.4469778 | 3.6647712 | -1.3263561 | 2.3318042 | -1.5551216 | 0.9925190 | -0.9818712 | -1.2474235 | -2.2427971 | 0.4255000 | -1.3293630 | 2.9865046 |
| -0.4190484 | 3.3717021 | -1.1904313 | -0.2723291 | -1.3349859 | 1.1433903 | -1.6972232 | 1.6679681 | -0.9818712 | -0.7312167 | -1.4575580 | 0.4167372 | -0.6725979 | 1.0511128 |
| -0.4188275 | 1.8710023 | -1.2953821 | -0.2723291 | -1.4299136 | 0.2396264 | -1.3028911 | 1.6679681 | -0.9818712 | -0.6422155 | -1.4575580 | 0.4167372 | -0.6193846 | -0.0579320 |
| -0.4178172 | 2.9429307 | -1.3668070 | -0.2723291 | -1.4644327 | -0.0777584 | -1.3171013 | 2.5141909 | -0.9818712 | -0.9922868 | -0.1180323 | -0.1651137 | 0.0387809 | -0.2645187 |
| -0.4156013 | 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | -0.1076467 | -1.3242064 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.4406159 | -0.7720229 | 0.0725439 |
| -0.4147654 | 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | 0.0432179 | -0.8161929 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.4406159 | -0.7076067 | -0.0253130 |
| -0.4151061 | 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | 0.3990304 | -1.6226198 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.1648063 | -0.4401395 | 0.2465117 |
| -0.4160303 | 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 0.8203125 | -1.4449928 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.3055571 | 0.6488122 |
| -0.4109336 | 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 1.2287853 | -1.4520979 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.2733490 | 1.6056352 |
| -0.4158989 | 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 0.4915417 | -1.6048571 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.1137088 | 0.5835743 |
| -0.4104929 | -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.2243532 | -1.7824842 | 0.8109650 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.5703722 | 0.1486548 |
| -0.4105697 | -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.3922967 | -0.9334268 | 0.8109650 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.3155082 | -0.0905509 |
| -0.4050672 | -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | 0.5598577 | -1.3313114 | 1.0283264 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.8938536 | 0.6596852 |
| -0.4138563 | -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | 0.3762584 | -0.6243557 | 1.0283264 | -0.6373311 | -0.7074831 | -1.1342242 | 0.3962541 | -0.7370141 | 0.4965904 |
| -0.4037058 | -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.7039885 | -0.3756778 | 1.1991001 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | 0.4462833 | -0.2427728 |
| -0.4125844 | 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 0.0859154 | -1.7220910 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.1266881 | -1.0758993 | -0.0035670 |
| -0.4136366 | 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 1.0764975 | -2.0808977 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.1632728 | -1.1081074 | 0.7031772 |
| -0.4149666 | 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 0.8345450 | -0.7522472 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.3744566 | -0.9218606 | 0.2465117 |
| -0.4159896 | 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.4346117 | -1.0009251 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.4280193 | -0.4415399 | -0.0579320 |
| -0.4093292 | 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.2994029 | -1.7824842 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.2950436 | -0.5577691 | 0.4204795 |
| -0.4084759 | 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.9925258 | -1.8073520 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.3697466 | -1.0913032 | 1.1489697 |
| -0.4136900 | 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 1.3540313 | -0.9760573 | 0.1077818 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4053456 | -0.8014303 | 1.4751593 |
| -0.4137319 | 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 0.4716162 | -0.3721252 | -0.2018524 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4018404 | -0.5213599 | 0.6379392 |
| -0.4113788 | 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 1.6159094 | 0.1181255 | -0.3304551 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4406159 | -0.8658465 | 1.1815886 |
| -0.4143678 | 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 0.8032335 | 0.0612849 | -0.2908010 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4406159 | -0.7174092 | 0.6161933 |
| -0.3627887 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4986579 | 0.4946949 | -0.2267846 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -1.1361145 | 0.0290519 |
| -0.3794811 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.4449570 | 0.2886475 | -0.3288880 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4333866 | -0.3757233 | -0.2427728 |
| -0.1137056 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -1.8667839 | -1.0932912 | -0.6058017 | -0.6373311 | -0.6184819 | -0.0256513 | -0.0681750 | -0.0018293 | -0.6994382 |
| -0.3282101 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.2314694 | -0.5604099 | -0.5483863 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -0.9344638 | -0.0470590 |
| -0.3896781 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.3723712 | 0.7753457 | -0.4563984 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4340438 | -0.1306617 | -0.3406296 |
| -0.3887841 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.0265214 | 0.5053525 | -0.2527616 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4021690 | -0.6655962 | -0.1014239 |
| -0.3771792 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4018769 | 0.6652169 | -0.0915333 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4273621 | -0.4723476 | 0.1377818 |
| -0.3906233 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.8249648 | 0.3241729 | 0.0712146 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4353582 | -0.1614694 | -0.6885653 |
| -0.3831002 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.5275055 | 0.5195627 | 0.0966692 | -0.6373311 | -0.6184819 | -0.0256513 | 0.3727041 | 0.7949710 | -0.5145975 |
| -0.3915928 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.7153745 | 0.1110204 | 0.1123884 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | 0.4602869 | -0.2971377 |
| -0.3733636 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.1385756 | -0.0488439 | -0.1246813 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4221044 | -0.3211096 | 0.0616709 |
| -0.3648244 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.2442787 | -0.3472574 | 0.0982364 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4332770 | 0.0107739 | -0.1666618 |
| -0.4006168 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.2011986 | -0.5781726 | 0.3539696 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.7636208 | 0.1377818 |
| -0.3989904 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.1300361 | -0.5071218 | 0.3539696 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.8098324 | 0.0616709 |
| -0.3792788 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.3467527 | -0.6634336 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.6936032 | -0.2318998 |
| -0.3870937 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.8206950 | 0.2033865 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.3774141 | -0.1278610 | -0.4384865 |
| -0.3804472 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.1855429 | -1.0115827 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.9148588 | 0.2682577 |
| -0.3977964 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.2083149 | -1.9139283 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4053456 | -1.0604955 | 0.2247657 |
| -0.3848208 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.0389481 | -1.4094674 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.9106578 | 0.0507979 |
| -0.3920800 | -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.2869762 | -0.8836912 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | 0.0191760 | -0.0361860 |
| -0.4124089 | -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.5929750 | -1.5195961 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2822280 | -0.3757233 | -0.3515026 |
| -0.4122845 | -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | 0.0688364 | -1.8251147 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2030341 | -0.7440159 | 0.0073060 |
| -0.4148189 | -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.2001579 | -1.2922335 | 0.9871052 | -0.6373311 | 0.1291279 | -0.7185093 | 0.1303027 | -0.4989543 | -0.2971377 |
| -0.4142620 | 1.0134596 | -0.7400171 | -0.2723291 | -1.0079168 | -0.8235415 | -1.4272301 | 1.3514003 | -0.9818712 | -0.6184819 | -0.7185093 | 0.4090698 | -0.0312367 | -0.5907084 |
| -0.4160722 | 1.0134596 | -0.7400171 | -0.2723291 | -1.0079168 | -0.3609852 | -1.6084097 | 1.3514003 | -0.9818712 | -0.6184819 | -0.7185093 | 0.0610765 | -0.6753986 | -0.3406296 |
| -0.4141923 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | 0.0446411 | -1.0826335 | 1.2648262 | -0.5224844 | -1.0931548 | 0.8057784 | 0.3618601 | -0.9764743 | -0.0361860 |
| -0.4157560 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | 0.0361016 | -1.0684234 | 1.2648262 | -0.5224844 | -1.0931548 | 0.8057784 | 0.3584645 | -0.8266367 | -0.1992808 |
| -0.4154967 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.3524457 | -1.2105250 | 1.0401514 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.6501923 | -0.1557889 |
| -0.4161175 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5915517 | -0.7913251 | 0.6819824 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.3995293 | -0.3297567 |
| -0.4165663 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5545472 | -0.3188371 | 0.8642962 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4177230 | -0.2931025 | -0.4384865 |
| -0.4162582 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.3211342 | -1.1110539 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4322912 | -0.5801747 | -0.2101538 |
| -0.4137110 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4264547 | -0.8232980 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4079314 | -0.3841216 |
| -0.4129506 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4506500 | -0.3579151 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4709472 | -0.4167406 |
| -0.4185892 | 1.0134596 | -1.4017907 | -0.2723291 | -0.9725347 | 1.3611476 | -0.6847489 | 1.5400303 | -0.9818712 | -0.7371501 | -1.3651769 | 0.4169563 | -1.0030810 | 1.1054777 |
| -0.4171976 | -0.4872402 | -1.3478576 | -0.2723291 | -0.3166707 | 0.3634492 | -0.3152846 | 1.1738830 | -0.9818712 | 0.0816606 | -1.1804147 | 0.3645985 | -0.5605698 | -0.6559463 |
| -0.4171452 | 1.8710023 | -1.0723615 | -0.2723291 | -0.6100835 | 0.5854762 | -0.4325184 | 0.9199069 | -0.5224844 | -0.2268768 | -0.3951756 | 0.4406159 | -0.7664215 | 0.1486548 |
| -0.4165570 | 1.8710023 | -1.0723615 | -0.2723291 | -0.6100835 | 0.8388147 | -1.4378877 | 1.2681505 | -0.5224844 | -0.2268768 | -0.3951756 | 0.3428010 | -1.1263120 | 0.9423829 |
| -0.4164826 | -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.3851804 | -0.7131692 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.3172793 | -0.2973036 | -0.5472164 |
| -0.4129379 | -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.5502775 | -0.5781726 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.0869268 | 0.0023717 | -0.5798354 |
| -0.4179277 | 3.1573164 | -1.0184285 | -0.2723291 | -1.0847220 | 0.3292912 | -1.4520979 | 2.2511443 | -0.6373311 | -0.3396116 | -0.2566040 | 0.3916537 | -0.8812504 | 0.0616709 |
| -0.4183566 | 2.9429307 | -1.3303658 | -0.2723291 | -1.0329432 | 0.4986579 | -1.3810470 | 2.1602961 | -0.6373311 | -0.7608838 | -0.6723188 | 0.3753329 | -0.9330634 | 0.2138927 |
| -0.4167314 | 1.2278453 | -1.4411472 | -0.2723291 | -1.0847220 | 0.9313260 | -1.2105250 | 2.3730984 | -0.9818712 | -0.4345461 | 0.5748257 | 0.3633936 | -0.9470669 | 0.4422255 |
| -0.4128809 | 1.2278453 | -1.4411472 | -0.2723291 | -1.0847220 | 0.2922867 | -0.8588234 | 2.3730984 | -0.9818712 | -0.4345461 | 0.5748257 | 0.4406159 | -0.9344638 | 0.0399249 |
| -0.4108592 | 2.0853880 | -1.3770106 | -0.2723291 | -1.2400582 | 0.4189559 | -1.1607894 | 3.2840500 | -0.6373311 | 0.0163931 | -0.0718418 | 0.1545100 | -1.0030810 | 0.1704008 |
| -0.4116799 | 2.0853880 | -1.3770106 | -0.2723291 | -1.2400582 | -0.5702030 | -1.7789317 | 3.2840500 | -0.6373311 | 0.0163931 | -0.0718418 | 0.3905583 | -0.6810000 | -0.4276135 |
| -0.4181148 | 3.3717021 | -1.3289081 | -0.2723291 | -1.2486880 | 0.6310202 | -1.1536844 | 3.9566022 | -0.5224844 | -1.3126910 | -0.6723188 | 0.3043541 | -1.1417159 | 0.8227800 |
| -0.4151014 | 2.9429307 | -1.3449423 | -0.2723291 | -1.2227986 | -0.8847413 | -1.6581453 | 3.2248775 | -0.6373311 | -0.4404796 | 1.6372081 | 0.2861713 | -0.6445909 | -0.4711055 |
| -0.4077097 | 2.9429307 | -1.3449423 | -0.2723291 | -1.2227986 | -0.4961940 | -1.7434063 | 3.2248775 | -0.6373311 | -0.4404796 | 1.6372081 | 0.2121255 | -0.9918782 | -0.2101538 |
| 0.6242409 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.1033769 | 1.0240236 | -0.7944316 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2306369 | 0.6927453 | -0.5145975 |
| 0.0274574 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | 0.1570779 | 0.7966610 | -0.6125452 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3797143 | 0.0863929 | -0.0905509 |
| 0.1846466 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.2243532 | 0.5266678 | -0.5092547 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4245142 | -0.1642701 | 0.0181789 |
| 0.0753105 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.2457019 | 0.4520644 | -0.6106931 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3731422 | 0.0023717 | 0.0073060 |
| 0.1079337 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | 0.1613476 | 0.6900847 | -0.6063715 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1959143 | -0.6810000 | 0.2682577 |
| 0.0259624 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.0478701 | 0.8002135 | -0.7121316 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0659843 | 0.2152253 | -0.2862647 |
| 0.0075215 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -1.3131396 | 0.9813931 | -0.8032647 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2641547 | -0.3449156 | -0.1884078 |
| 0.0707857 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.6854862 | 0.7256101 | -0.8977222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0398054 | 0.2782411 | -0.6233273 |
| -0.0161882 | -0.4872402 | 1.0149946 | 3.6647712 | 1.4092873 | 3.5515296 | 0.5089051 | -0.8977222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0232656 | -1.0310881 | -0.0688050 |
| 0.1095555 | -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -3.8764132 | 0.6865322 | -1.0361553 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0216226 | -0.7748236 | 0.5400824 |
| 0.0096990 | -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -1.8810164 | 0.8108711 | -0.9700968 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4451952 | 0.1886186 | -0.0688050 |
| 1.1519647 | -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -3.4465917 | 1.1163897 | -1.0848799 | 1.6596029 | 1.5294129 | 0.8057784 | -2.4673242 | 0.0947950 | 0.0616709 |
| 0.1493565 | -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -1.8710537 | 1.1163897 | -1.1694595 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2064297 | -1.3153595 | 2.9865046 |
| 0.2390799 | -0.4872402 | 1.0149946 | 3.6647712 | 0.6584956 | 0.5669739 | 1.0027084 | -1.1579669 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2043485 | -1.2495430 | 2.9865046 |
| 0.3400827 | -0.4872402 | 1.0149946 | 3.6647712 | 0.6584956 | 1.0409163 | 1.0275762 | -1.2312439 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3874913 | -1.3573701 | 2.9865046 |
| 0.6532287 | -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -0.0976839 | 1.1163897 | -1.2470580 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1037952 | -0.4373388 | 2.9865046 |
| 0.5410338 | -0.4872402 | 1.0149946 | 3.6647712 | 0.9777978 | -0.5830122 | 0.7469254 | -1.2658165 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0963256 | -0.5283617 | 2.9865046 |
| 0.8713058 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -1.9621417 | 1.1163897 | -1.2446360 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.0971497 | -0.9495170 |
| 1.7304654 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -3.0551979 | 1.1163897 | -1.2623023 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.5452624 | -0.9495170 |
| 1.8596166 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 1.4636216 | 1.0417863 | -1.1771529 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.1101988 | -0.8190411 |
| 1.3572534 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.5185834 | 0.8783694 | -1.1635707 | 1.6596029 | 1.5294129 | 0.8057784 | 0.0695107 | 1.4825438 | -0.9386440 |
| 0.7219594 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.7249547 | 1.0737592 | -1.1573496 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.2024734 | -1.0038819 |
| 2.3291951 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.1357291 | 0.9813931 | -1.1440049 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.5455597 | -1.0256279 |
| 1.6570484 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | -0.0877212 | 1.1163897 | -1.1440049 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4060028 | 1.2780924 | -1.3409445 |
| 9.9241096 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.9726003 | 0.8286338 | -1.1295680 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.6381316 | -1.3191985 |
| 1.4254272 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.3705654 | 1.0844168 | -1.0807958 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.1800678 | -1.2648336 |
| 0.6479646 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.0654941 | 1.1163897 | -1.0517320 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.5329565 | -1.2213417 |
| 0.5090895 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.0882661 | 1.1163897 | -1.0741947 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.6673903 | -1.1126118 |
| 1.9149323 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -2.7278503 | 0.8037660 | -1.1186453 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7759914 | 2.5174040 | -1.4931663 |
| 1.5344076 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.4341159 | 1.0488914 | -1.1250089 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.5426103 | -1.6671342 |
| 2.4158772 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -2.3236472 | 1.1163897 | -1.1054906 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.1883213 | -1.3083256 |
| 2.2069961 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.8283562 | 0.7433728 | -1.0811757 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.7078519 | -1.6453882 |
| 1.2463082 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.9991462 | 1.1163897 | -1.0474104 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1779505 | 2.5160036 | -1.3409445 |
| 0.5276048 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.2732886 | 1.0773117 | -0.9815894 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.1478597 | -1.1995957 |
| 0.3893052 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -0.8135788 | 1.0098134 | -0.8873694 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4135607 | 0.6241280 | -0.8081681 |
| 0.1952587 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -0.3325202 | 0.4946949 | -0.7727762 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2377567 | 0.8551861 | 0.0725439 |
| 0.9259239 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.7771192 | 1.0098134 | -0.9616911 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.8242297 | -1.3953095 |
| 0.5849224 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.1304187 | 0.8535016 | -0.9516232 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.3524598 | -0.9495170 |
| 1.1330844 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.5659332 | 0.9281050 | -0.9559448 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5177013 | -1.0691198 |
| 0.5932918 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.2652449 | 1.0737592 | -0.9827291 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3867246 | 0.6255284 | -1.0256279 |
| 0.2625722 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1713104 | 0.9742880 | -1.0059517 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.9406076 | -1.0908658 |
| 0.4718334 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.7651883 | 1.0773117 | -1.0265623 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3989925 | 1.0176270 | -1.5257853 |
| 4.0386089 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.1836238 | 1.1163897 | -1.0948528 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.5118026 | -1.9063399 |
| 0.7327784 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.6157470 | 0.3277255 | -1.0897239 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2027938 | 2.4249808 | -1.7649910 |
| 2.4917124 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.4236082 | 1.1163897 | -1.0477428 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.9768681 | -1.8411020 |
| 1.2349731 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.0830689 | 1.1163897 | -1.0547238 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.0736410 | -1.6671342 |
| 0.6954781 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1698871 | 1.1163897 | -1.0239029 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2128922 | 1.0722407 | -1.1343578 |
| 2.4632989 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.3316418 | 0.9742880 | -0.9936043 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.9966217 | -1.5475313 |
| 4.4080076 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.0726103 | 0.5977186 | -1.0389097 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2980894 | 2.0622896 | -1.5257853 |
| 7.4762471 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.8562763 | 1.1163897 | -1.1253414 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3099404 | 1.4461347 | -1.9063399 |
| 1.9883261 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9001297 | -3.0551979 | 1.1163897 | -1.2427839 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1483760 | 1.4965474 | -1.1561037 |
| 0.9693115 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9001297 | -0.9630200 | 1.1163897 | -1.1919222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2692816 | -0.0732473 | 0.5835743 |
| 0.4406611 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -0.9502108 | 1.0417863 | -1.1114268 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4604205 | 1.9250551 | -0.5798354 |
| 1.2584688 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | 0.8075032 | 1.1163897 | -1.1062979 | 1.6596029 | 1.5294129 | 0.8057784 | -1.9422126 | 0.9980220 | 0.5400824 |
| 5.5248535 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -0.7509558 | 1.1163897 | -1.1312301 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8783565 | -0.3561184 | -0.8190411 |
| 1.2134072 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | 0.5299694 | 1.1163897 | -1.0768541 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5229148 | 1.1996727 | -0.5798354 |
| 1.7668310 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -2.3578052 | 1.1163897 | -1.0643168 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5914839 | 3.0411357 | -0.5037245 |
| 2.9113695 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -1.6077524 | 1.1163897 | -1.0474579 | 1.6596029 | 1.5294129 | 0.8057784 | -1.5959718 | 1.0400326 | -0.6776923 |
| 4.8982568 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -2.5129395 | 1.1163897 | -1.0147848 | 1.6596029 | 1.5294129 | 0.8057784 | -2.9399686 | 3.4066275 | -1.6888801 |
| 1.6823810 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.2125846 | 1.1163897 | -0.9309651 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6083523 | 2.2961484 | -1.6671342 |
| 0.8394627 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.7078757 | 0.7895559 | -0.9381836 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6705683 | 1.8396336 | -1.6345152 |
| 2.5957053 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -1.3956881 | 0.7291627 | -1.0198662 | 1.6596029 | 1.5294129 | 0.8057784 | -2.5117955 | 1.9586635 | -1.3191985 |
| 8.1288391 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.4663057 | 1.1163897 | -0.9462094 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7266503 | 1.1156516 | -1.4931663 |
| 0.9531748 | -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | 0.7676522 | 0.2815424 | -0.9502935 | 1.6596029 | 1.5294129 | 0.8057784 | -3.3761377 | 1.4125262 | -1.5366583 |
| 0.8688993 | -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | 0.1798499 | 1.1163897 | -0.9194726 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4154016 | 0.3314545 | -0.6342003 |
| 0.3963319 | -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -0.3965665 | 0.9494202 | -0.9120166 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4019288 | 0.4266784 | -0.9060250 |
| 0.9806002 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.9060900 | 0.6758745 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7133373 | 0.2026221 | -0.1884078 |
| 0.3995673 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.2585112 | 0.5870610 | -0.8421115 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8792328 | 1.4895456 | -0.9930089 |
| 0.6020542 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -1.0242198 | 0.0719425 | -0.8223082 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8668553 | 0.6311298 | -1.1778497 |
| 1.4237880 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.5531240 | 0.9529728 | -0.8953952 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8227126 | 1.6435843 | -1.5475313 |
| 1.0037355 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.6370957 | -0.3152846 | -0.8536040 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6368314 | 0.4252781 | -1.3409445 |
| 3.9584024 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.1176094 | 0.3596983 | -0.9175730 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7006904 | 0.2614369 | -1.2648336 |
| 0.4363851 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.1304187 | 0.3383831 | -0.8830478 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8473018 | 1.2416833 | -1.2539606 |
| 0.6656207 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.1357291 | 0.9600779 | -0.8675661 | 1.6596029 | 1.5294129 | 0.8057784 | -3.2417380 | 1.6001734 | -1.4170554 |
| 0.5671779 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.0901851 | 0.6225864 | -0.8274371 | 1.6596029 | 1.5294129 | 0.8057784 | -2.9927645 | 0.6983467 | -0.8734060 |
| 0.7497230 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.7804615 | 0.9138948 | -0.8105782 | 1.6596029 | 1.5294129 | 0.8057784 | -3.0159860 | 0.9854189 | -0.9168980 |
| 0.3290719 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.1997754 | 0.2211492 | -0.7572945 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8339385 | -0.0872508 | -0.6994382 |
| 0.2287434 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.2154311 | 0.6865322 | -0.7024911 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8094026 | 0.4994967 | -0.8951520 |
| 1.1974449 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1090699 | 0.9387626 | -0.7469417 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8045831 | 0.3524598 | -1.1778497 |
| 0.8773861 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.4901184 | 0.9245525 | -0.7932444 | 1.6596029 | 1.5294129 | 0.8057784 | -2.7035916 | 1.4867449 | -0.9930089 |
| 1.2564343 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2510124 | 0.8783694 | -0.8512296 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6057234 | 0.7557611 | -1.4061824 |
| 1.3443720 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.1887719 | 1.1163897 | -0.8932106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8047489 | 1.9320568 | -1.5040393 |
| 1.1700894 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.4976172 | 0.6865322 | -0.9376612 | 1.6596029 | 1.5294129 | 0.8057784 | -3.1515905 | 2.9921233 | -1.5366583 |
| 0.6716359 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.9359783 | 0.8996847 | -0.9392759 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.4321312 | -1.0582468 |
| 2.1435191 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.6641375 | 0.8463965 | -0.9160058 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3809192 | 1.3243041 | -1.3083256 |
| 0.7104138 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.1727336 | 1.0169185 | -0.8215484 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3207844 | 0.9616129 | -0.5907084 |
| 0.2386602 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.0934142 | 1.1163897 | -0.8501848 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4273621 | 0.5513097 | -0.4493595 |
| 0.7385901 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2851704 | 1.1163897 | -0.8627221 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3292186 | 0.8677893 | -0.7755492 |
| 1.0682704 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.6129005 | 0.9956033 | -0.9020438 | 1.6596029 | 1.5294129 | 0.8057784 | -1.2722954 | 1.5595632 | -1.2757066 |
| 0.8205824 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2481659 | 0.9316575 | -0.8582106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.4351771 | 1.5861699 | -1.1669767 |
| 0.3109379 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.0802224 | 0.9884982 | -0.8182715 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4235072 | 0.7193520 | -0.8299141 |
| 0.7337433 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.0478701 | 0.9956033 | -0.7584343 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3488254 | 0.5303045 | -1.0799928 |
| 0.6644814 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1418047 | 1.0702067 | -0.7282307 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.7669640 | -0.9168980 |
| 0.4548586 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.1883894 | 1.0559965 | -0.7646079 | 1.6596029 | 1.5294129 | 0.8057784 | -0.5746657 | 0.9322055 | -1.0365009 |
| 0.3608882 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.6609085 | 0.8535016 | -0.6987869 | 1.6596029 | 1.5294129 | 0.8057784 | -3.9033305 | 0.6703397 | -0.9930089 |
| 0.2124754 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5271229 | 1.0524439 | -0.6837801 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0151600 | 0.7109499 | -0.7972951 |
| 0.1716722 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0175994 | 0.8250813 | -0.6776064 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3112548 | 0.6465337 | -0.6994382 |
| 0.5388063 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 1.5774816 | 1.0915219 | -0.6374774 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2102634 | 0.5723150 | -0.5145975 |
| 0.6859357 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.6310202 | 0.9067897 | -0.6168668 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8336662 | 0.8481844 | -0.8299141 |
| 0.1324002 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.3421004 | 0.6367966 | -0.6455032 | 1.6596029 | 1.5294129 | 0.8057784 | -3.3490825 | 0.7669640 | -0.9168980 |
| 0.1226880 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4392640 | 0.6865322 | -0.5767378 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7920428 | 0.8901949 | -1.0691198 |
| 0.5332828 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4961940 | 0.4165390 | -0.4824229 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8684983 | 0.6003221 | -0.9821359 |
| 0.4811585 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0232924 | 0.5373254 | -0.4805707 | 1.6596029 | 1.5294129 | 0.8057784 | -0.9251783 | 0.5008971 | -0.8299141 |
| 0.3705900 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.2898227 | 0.5621932 | -0.5117241 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.2866432 | -0.2753917 |
| 0.1393478 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5925924 | 0.7611355 | -0.5687120 | 1.6596029 | 1.5294129 | 0.8057784 | -1.1111691 | 0.5275038 | -0.6668193 |
| 0.0092526 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.1300361 | 0.7042949 | -0.5831490 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3807001 | 0.2796414 | -0.5254704 |
| 0.3535872 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0460644 | 0.5124576 | -0.5036983 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.1872182 | -0.3297567 |
| 0.2566546 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.3250214 | 0.7575830 | -0.4717851 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4068791 | -0.3309120 | -0.2536457 |
| 0.4912834 | -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.1076467 | -0.1127897 | -0.3949464 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.0793911 | -0.1231699 |
| -0.0523073 | -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.7481093 | -0.7238268 | -0.3459843 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2439790 | 0.2068231 | -0.2862647 |
| 0.0187706 | -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.4734220 | 0.5728508 | -0.4385897 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6657487 | 0.6297295 | -0.3841216 |
| 0.0940246 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.4008362 | 0.9209999 | -0.5958763 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2780445 | 1.2136763 | -0.3732486 |
| 1.3907009 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.5104265 | 0.0861526 | -0.4210659 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1321648 | 0.7669640 | -0.3732486 |
| 1.0999857 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.8135788 | -0.4218608 | -0.4612898 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.2950453 | -0.2645187 |
| 0.0854807 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.1674232 | 0.5479830 | -0.3617035 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5092992 | -0.2862647 |
| 0.0493965 | -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.0791817 | 0.7860033 | -0.3304076 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4234189 | 0.0303788 | -0.3188837 |
| -0.0052134 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | 0.2168544 | 0.2282543 | -0.4267172 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4019500 | 0.2390312 | 0.0725439 |
| 0.1201373 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | 0.9896793 | -0.0346338 | -0.5993905 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1972287 | -0.1390638 | 0.7901611 |
| 0.5164498 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -1.2206283 | 0.9529728 | -0.6483526 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0448441 | 0.7683643 | -0.9495170 |
| 0.3231508 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.1745394 | 1.0240236 | -0.7546351 | 1.6596029 | 1.5294129 | 0.8057784 | -0.5905484 | 1.6029741 | -1.0038819 |
| 0.1462396 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | 0.2837472 | 0.8890270 | -0.7074776 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4330580 | 0.8439833 | -0.6342003 |
| 1.3264915 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -1.3956881 | 1.0204711 | -0.8046419 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0788000 | 1.7164026 | -1.1452307 |
| 0.7695683 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.1418047 | 0.9991558 | -0.7714940 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2522154 | 0.7529604 | -0.8625331 |
| 1.2463082 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.0791817 | 0.6900847 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2918671 | 0.0639872 | -0.1231699 |
| 0.2569871 | -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.0606794 | -0.1376575 | -0.1761129 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | -0.2678962 | 0.0507979 |
| 0.2435210 | -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | 0.6623317 | 0.2247018 | -0.2200411 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3986639 | -0.6880018 | 0.1269088 |
| 0.2461926 | -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | 1.1049625 | 0.2993051 | -0.1825715 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4228712 | -0.7902275 | 0.2682577 |
| -0.0924419 | -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.7438395 | -1.0044776 | 0.1440166 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3970209 | -0.3127075 | -0.0796779 |
| -0.1435735 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5887052 | -0.9476370 | -0.0337381 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1539623 | 0.0961953 | -0.2101538 |
| 0.0069925 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | 0.0389481 | -0.5923828 | 0.0933924 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3499208 | -0.2903018 | -0.1449159 |
| 0.2416108 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.2428554 | 0.3987763 | -0.1183177 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3943920 | 0.3258531 | -0.3732486 |
| 0.1420845 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5403147 | -0.5461998 | -0.3052380 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3455394 | -0.1684712 | -0.2101538 |
| -0.4025630 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.1822006 | 0.8570542 | -0.9375187 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4207900 | 0.7571615 | -0.7972951 |
| -0.3987834 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.2391306 | 1.0559965 | -0.9686246 | -0.6373311 | 1.7964164 | 0.7595879 | -0.1382776 | 1.5847695 | -1.6888801 |
| -0.3959828 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.6959939 | 1.0453389 | -0.9367114 | -0.6373311 | 1.7964164 | 0.7595879 | -0.4189067 | 2.3843705 | -1.5692773 |
| -0.4078085 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -0.4293012 | 1.0737592 | -0.9151035 | -0.6373311 | 1.7964164 | 0.7595879 | 0.3662415 | 0.7585618 | -0.9712629 |
| -0.4071598 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -0.4293012 | 0.5302203 | -0.8002729 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4406159 | 0.0975957 | -0.2645187 |
| -0.3999530 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.8221183 | -0.5177794 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | -0.0900515 | -0.0796779 |
| -0.3875994 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.5104265 | -0.9227692 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.1312041 | 0.2138927 |
| -0.3992926 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.8747785 | -1.4130199 | -0.4732098 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4010737 | 0.6927453 | 0.0616709 |
| -0.3864333 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -1.2732886 | 0.1536509 | -0.4732098 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 1.1884699 | -0.3080107 |
| -0.3889003 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.6982955 | 0.0719425 | -0.4285218 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.2026221 | -0.4602325 |
| -0.3923020 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.3780642 | -0.1163422 | -0.6581830 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.0373805 | -0.1449159 |
| -0.3994275 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -1.0185268 | 0.1749662 | -0.6625521 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4282384 | 0.3426573 | -0.5472164 |
| -0.3940157 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.3666782 | 0.3952238 | -0.6158695 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.2348302 | -0.6233273 |
| -0.4128204 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.4388814 | 0.0186544 | -0.6251775 | -0.9818712 | -0.8024176 | 1.1753027 | 0.3868341 | -0.4177339 | -0.0144400 |
| -0.4148387 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | -0.2343159 | 0.2886475 | -0.7159308 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.5003546 | -0.2101538 |
| -0.4130378 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.9839863 | 0.7966610 | -0.7729187 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.9820757 | 0.1486548 |
| -0.4073609 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.7249547 | 0.7362677 | -0.6677760 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4028263 | -0.8644462 | -0.0579320 |
| -0.4145899 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | -0.3624084 | 0.4343017 | -0.6126402 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.6683969 | -1.1561037 |
1.3.2. Create a categorical variable (crime rate)
# summary of the scaled crime rate
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
knitr::kable(bins, caption="Quantiles of crim") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| x | |
|---|---|
| 0% | -0.4193669 |
| 25% | -0.4105633 |
| 50% | -0.3902803 |
| 75% | 0.0073892 |
| 100% | 9.9241096 |
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
# look at the table of the new factor crime
knitr::kable(table(crime), caption="Categorical variables of crime") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| crime | Freq |
|---|---|
| low | 127 |
| med_low | 126 |
| med_high | 126 |
| high | 127 |
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
knitr::kable(boston_scaled, caption = "Scaled Boston data set with caterogical variable crime") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center",) %>%
scroll_box(width = "100%", height = "300px") # the data frame head| zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | crime |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.2845483 | -1.2866362 | -0.2723291 | -0.1440749 | 0.4132629 | -0.1198948 | 0.1400750 | -0.9818712 | -0.6659492 | -1.4575580 | 0.4406159 | -1.0744990 | 0.1595278 | low |
| -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 0.1940824 | 0.3668034 | 0.5566090 | -0.8670245 | -0.9863534 | -0.3027945 | 0.4406159 | -0.4919525 | -0.1014239 | low |
| -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 1.2814456 | -0.2655490 | 0.5566090 | -0.8670245 | -0.9863534 | -0.3027945 | 0.3960351 | -1.2075324 | 1.3229375 | low |
| -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.0152978 | -0.8090878 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4157514 | -1.3601708 | 1.1815886 | low |
| -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.2273620 | -0.5106743 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4406159 | -1.0254866 | 1.4860323 | low |
| -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 0.2068916 | -0.3508100 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4101651 | -1.0422909 | 0.6705582 | low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3880270 | -0.0701592 | 0.8384142 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4263763 | -0.0312367 | 0.0399249 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.1603069 | 0.9778406 | 1.0236249 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4406159 | 0.9097999 | 0.4965904 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.9302853 | 1.1163897 | 1.0861216 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3281233 | 2.4193794 | -0.6559463 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3994130 | 0.6154813 | 1.3283202 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3289995 | 0.6227277 | -0.3949946 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | 0.1314594 | 0.9138948 | 1.2117800 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3926395 | 1.0918456 | -0.8190411 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3922967 | 0.5089051 | 1.1547920 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4406159 | 0.0863929 | -0.3949946 | med_low |
| 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.5630867 | -1.0506607 | 0.7863653 | -0.5224844 | -0.5769480 | -1.5037485 | 0.3705134 | 0.4280788 | -0.0905509 | med_low |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4776917 | -0.2406812 | 0.4333252 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4406159 | -0.6151835 | -0.2318998 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2684739 | 0.5657458 | 0.3166900 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2557205 | -0.3351131 | -0.4711055 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6413655 | -0.4289659 | 0.3341188 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4265954 | -0.5857761 | -0.2862647 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4976172 | -1.3952572 | 0.3341188 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3305330 | -0.8504426 | 0.0616709 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4193385 | 0.4662746 | 0.2198106 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3294377 | 0.2824421 | -0.5472164 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.1793541 | -1.1359217 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | -0.7413783 | -0.1348628 | -0.2536457 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.7936533 | 0.0328645 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3754425 | -0.1922772 | -0.4711055 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.0171035 | 1.0488914 | 0.0013569 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2179309 | 1.1716657 | -0.9712629 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4549197 | 0.7327152 | 0.1031753 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3927490 | 0.1648126 | -0.3188837 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2030044 | 0.8215287 | 0.0863639 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4406159 | 0.8495847 | -0.7972951 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 1.1163897 | 0.1425445 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4147656 | 1.0120256 | -0.8734060 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.5132730 | 0.9067897 | 0.2871038 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4124654 | 0.5106995 | -0.7538032 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.9758293 | 0.6083763 | 0.3132232 | -0.6373311 | -0.6006817 | 1.1753027 | -0.5833190 | 0.5401069 | -0.9386440 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 0.7717932 | 0.4212153 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2213265 | 0.3020471 | -0.6450733 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.3382132 | 0.7185050 | 0.3126533 | -0.6373311 | -0.6006817 | 1.1753027 | -0.5508966 | 0.6479340 | -0.8407871 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | 0.2994029 | 0.9174474 | 0.3132707 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3424724 | 0.0205763 | -0.4493595 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | 0.5541647 | 0.6652169 | 0.2108350 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2580207 | -0.0942525 | -0.1666618 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8135788 | 0.9067897 | 0.2079856 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0382932 | 1.3929213 | -1.0691198 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.3026319 | 1.1163897 | 0.1804414 | -0.6373311 | -0.6006817 | 1.1753027 | 0.2196834 | 0.0541848 | -0.8734060 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.4762685 | 0.4769322 | 0.0925851 | -0.6373311 | -0.6006817 | 1.1753027 | -1.3590472 | 2.1085012 | -1.0147549 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8306578 | 0.9387626 | -0.0037245 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0229582 | 0.7977717 | -1.0256279 | med_high |
| -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.2684739 | 1.0062609 | -0.0167367 | -0.6373311 | -0.6006817 | 1.1753027 | -1.1869674 | 1.0764418 | -0.9821359 | med_high |
| -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.5004637 | -0.0133185 | -0.2064589 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.4163335 | -0.3949946 | low |
| -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6314027 | -0.2548913 | -0.1981007 | -0.5224844 | -0.7668172 | 0.3438730 | 0.2287748 | -0.1740726 | -0.2753917 | med_low |
| -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6185935 | -0.9618471 | 0.0660857 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.5437656 | -0.1666618 | low |
| -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.4534965 | -1.3632843 | 0.0248170 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4026072 | -0.3533177 | 0.2356387 | med_low |
| 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 0.4417279 | -1.6616978 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4267049 | -1.1669222 | 0.8988910 | low |
| 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 1.0523023 | -1.8748503 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4265954 | -1.4946046 | 1.3446835 | low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | 0.6907967 | -2.3331282 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3147600 | -1.0941039 | 0.4422255 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1645767 | -2.2016841 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.2924148 | -0.9582698 | 0.3008766 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1048002 | -2.2052367 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4138893 | -0.7300124 | 0.2356387 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.3069017 | -1.0151352 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3583550 | -0.4345381 | -0.1449159 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.8576995 | -1.2353928 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | -0.3421149 | -0.3515026 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.7096815 | -1.2531555 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.2096238 | -0.2753917 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.3624084 | 0.6012712 | 0.8996287 | -0.7521778 | -1.0397541 | -0.2566040 | 0.3950493 | 0.8607875 | -0.6450733 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -1.2604793 | 0.9494202 | 0.9853955 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 2.5426103 | -0.8842790 | med_low |
| -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.9715595 | -0.2335761 | 1.0887811 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.4966960 | -0.3406296 | med_low |
| 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4577662 | -0.8126404 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4259382 | 0.1115992 | -0.3080107 | med_low |
| 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.2414322 | -0.1980507 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4085221 | -0.4513423 | -0.2210268 | low |
| 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | 0.3221749 | -1.6865656 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -1.0324884 | 0.2682577 | low |
| 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4079525 | -1.6759080 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -0.5913775 | 0.0942899 | low |
| 2.7285450 | -1.0402932 | -0.2723291 | -1.2486880 | -0.5645100 | -0.7451421 | 1.6738568 | -0.7521778 | 0.3605309 | 1.2214933 | 0.4406159 | 0.3006467 | -0.3949946 | low |
| 3.3717021 | -1.4455202 | -0.2723291 | -1.3090965 | 1.3725336 | -1.6581453 | 2.3277455 | -0.5224844 | -1.0812880 | -0.2566040 | 0.4299910 | -1.0983050 | 1.3990484 | low |
| 3.1573164 | -1.5154874 | -0.2723291 | -1.2486880 | 0.1399989 | -1.1678945 | 2.5609210 | -0.8670245 | -0.5650812 | -0.5337472 | 0.4406159 | -0.9638712 | 0.2356387 | low |
| 3.8004735 | -1.4309437 | -0.2723291 | -1.2400582 | 0.7562662 | -0.9973725 | 2.1511780 | -0.5224844 | -0.9032856 | -1.5499390 | 0.3968018 | -1.2187352 | 0.9858749 | low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.1987347 | -1.3988097 | 1.9089794 | -0.1779443 | -0.7371501 | 0.5748257 | 0.3724850 | -0.8112328 | 0.0834169 | med_low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.5090032 | -0.7593522 | 1.4897384 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.4807497 | -0.3188837 | med_low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.7737278 | -0.0843694 | 1.6290739 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4210091 | 0.0695886 | -0.4167406 | med_low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.4534965 | 0.8819220 | 1.4358374 | -0.1779443 | -0.7371501 | 0.5748257 | 0.2344707 | 0.2502341 | -0.7103112 | med_low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.2438961 | -0.0275287 | 1.6291213 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.8294374 | -0.0361860 | med_low |
| 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.6794107 | -0.8943488 | 1.9878602 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4261573 | -0.4415399 | 0.2682577 | med_low |
| 0.2631097 | -1.4221978 | -0.2723291 | -1.1960462 | 1.1661623 | -0.3223896 | 2.5776850 | -0.7521778 | -1.1406221 | 0.0667298 | 0.4005260 | -0.6445909 | 1.1380967 | low |
| 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | 0.0076366 | -1.8037995 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -1.1179099 | 0.1051628 | low |
| 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | -0.7082582 | -1.3313114 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -0.3379138 | -0.3406296 | low |
| 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5787425 | -1.6759080 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4330580 | -0.6375891 | -0.0579320 | low |
| 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.9829455 | -1.1288166 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | 0.0611865 | -0.5580894 | med_low |
| 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5687797 | -1.2638131 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | -0.5409648 | -0.1775348 | med_low |
| -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | 0.1883894 | -2.2016841 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2963581 | -0.8308377 | 0.1812738 | med_low |
| -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.4606127 | -1.8144571 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2219837 | -0.3883265 | -0.0905509 | med_low |
| -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.3125947 | -2.1590536 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.3750043 | -0.9988800 | 0.0290519 | med_low |
| -0.4872402 | -0.0476329 | -0.2723291 | -1.2227986 | -0.0564097 | -2.2158943 | 0.7086718 | -0.6373311 | -0.6125485 | 0.3438730 | 0.2245030 | -0.7160089 | 0.0942899 | med_low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0165586 | -2.2229994 | 0.2167712 | -0.5224844 | -0.0607412 | 0.1129203 | 0.4189279 | -0.8224356 | 0.1704008 | low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | 0.0019436 | -0.8375082 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.2908813 | -0.5199596 | -0.1231699 | med_low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0080191 | 0.2104916 | 0.1221238 | -0.5224844 | -0.0607412 | 0.1129203 | 0.1860561 | -0.0956529 | -0.2753917 | med_low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.2058509 | -0.8090878 | 0.1403124 | -0.5224844 | -0.0607412 | 0.1129203 | 0.3317379 | -0.3337127 | -0.1884078 | med_low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.0749119 | -0.5284370 | 0.5789293 | -0.5224844 | -0.0607412 | 0.1129203 | 0.3256039 | -0.0438399 | -0.1449159 | low |
| -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | -0.5844355 | -1.1359217 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.4314149 | -0.4975539 | -0.2427728 | med_low |
| 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.6295970 | -1.2460504 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4406159 | -1.0310881 | 0.5944473 | low |
| 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.4758859 | 0.0648374 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4267049 | -0.7608201 | 0.1486548 | low |
| 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | 0.0247156 | -1.2922335 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.4406159 | -0.8308377 | 0.2465117 | low |
| 0.5846882 | -0.9149352 | -0.2723291 | -1.1106113 | -0.1674232 | -0.7771150 | 0.7625253 | -0.6373311 | -0.7549503 | 0.2514920 | 0.3720469 | -0.7202099 | 0.0399249 | low |
| -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.1485384 | -0.7309319 | 0.4674705 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4406159 | -0.4247356 | 0.1486548 | low |
| -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.4915417 | -0.4431760 | 0.3051974 | -0.7521778 | -0.9566863 | 0.0205393 | 0.3902297 | -0.8574444 | 0.4422255 | low |
| -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | -0.3837572 | -0.8339556 | 0.3002110 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4306482 | 0.0289784 | -0.0035670 | low |
| -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | -0.2328927 | -0.4183083 | -0.0225305 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4214472 | -0.5899772 | -0.0361860 | low |
| -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 1.0281070 | 0.6296915 | -0.1773001 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4406159 | -1.0016807 | 0.1160358 | low |
| -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 1.1305810 | -0.1944981 | -0.1807194 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4314149 | -0.9736736 | 0.6705582 | low |
| -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 0.1883894 | -0.0879219 | -0.3337319 | -0.8670245 | -0.8202179 | -0.3027945 | 0.3889153 | -0.5381641 | 0.0073060 | low |
| -0.4872402 | -1.1262946 | -0.2723291 | -0.5669346 | 0.1713104 | 0.1891763 | -0.3338269 | -0.8670245 | -0.8202179 | -0.3027945 | 0.4039216 | -0.6235856 | -0.0579320 | low |
| 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | 0.2239706 | -0.5319896 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4199137 | -0.6291870 | 0.0399249 | low |
| 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.1048002 | -1.4094674 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4343724 | -0.9022557 | 0.2682577 | low |
| 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.0507166 | 0.3099628 | -0.0855021 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4406159 | -0.2889015 | -0.2101538 | low |
| -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 0.4844254 | -0.3827828 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.0143049 | -0.8406402 | 0.6379392 | med_low |
| -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | -0.1731162 | 0.0364171 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.3850816 | -0.1838751 | -0.1231699 | med_low |
| -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 2.5395987 | 0.2637797 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4406159 | -1.1823261 | 1.7578570 | med_low |
| -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 2.1852094 | -1.1252640 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4037025 | -1.2719486 | 2.3123794 | low |
| -0.4872402 | -1.2020925 | -0.2723291 | -0.9466453 | 1.6102164 | -0.2158134 | -0.1423950 | -0.8670245 | -0.7846174 | -0.2104134 | 0.4406159 | -0.9050564 | 1.1598427 | low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.6295970 | 0.4023288 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4171754 | -0.4527427 | 0.5400824 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.7064525 | 0.0968103 | -0.4459031 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4261573 | -0.6978043 | 0.4313525 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.1713104 | 0.5977186 | -0.5130539 | -0.5224844 | -0.1438090 | 1.1291122 | -3.1313265 | -0.2833001 | -0.4276135 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.2101207 | 0.6687695 | -0.5130539 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4139988 | 0.1101988 | -0.3515026 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1674232 | 0.7611355 | -0.6525317 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3945016 | -0.0452402 | -0.2645187 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6171702 | 0.9991558 | -0.8016976 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4093984 | 0.5345055 | -0.3297567 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6385190 | 0.8286338 | -0.7522606 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4271431 | 0.8411826 | -0.3297567 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.2243532 | 0.5906135 | -0.7943366 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3397340 | 0.2012217 | -0.2318998 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | 0.2695146 | 1.0133660 | -0.6468804 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4224331 | -0.0536423 | -0.2971377 | med_low |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.0791817 | 0.8037660 | -0.5935968 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3785094 | 0.4056731 | -0.3406296 | med_high |
| -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1275722 | -0.5035693 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4032644 | 0.0485834 | -0.0905509 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | 0.6125180 | 0.4627220 | -0.5307201 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4262668 | -0.3491166 | 0.0290519 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5289287 | 0.8641592 | -0.6846349 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4192565 | 0.4980964 | -0.4058676 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.2741669 | 0.9529728 | -0.5922195 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4406159 | 0.6213273 | -0.4167406 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.0436004 | 0.5550881 | -0.7306527 | -0.4076377 | 0.1409947 | -0.3027945 | 0.3512352 | -0.3085064 | -0.4384865 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5075800 | 0.6971898 | -0.6325385 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1288575 | 0.4350805 | -0.4602325 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.1546139 | 0.1394408 | -0.5057404 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4011832 | -0.0858504 | -0.1449159 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.3752177 | 0.4982475 | -0.4975246 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4144370 | -0.3295117 | -0.3623756 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5872820 | 0.1607560 | -0.6256999 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1976456 | 0.3804668 | -0.2318998 | med_low |
| -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.7879603 | -0.1198948 | -0.4919208 | -0.4076377 | 0.1409947 | -0.3027945 | 0.3814669 | 0.1340048 | -0.3515026 | med_low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.5901285 | 0.0399696 | -0.7300828 | -0.8670245 | -1.3067576 | 0.2976825 | 0.3557261 | 0.2404316 | -0.0579320 | low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.3994130 | 0.5515356 | -0.7587192 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2299797 | 0.2264281 | -0.2427728 | low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.4606127 | 0.8641592 | -0.8111956 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2345802 | 0.7389569 | -0.2210268 | med_low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.6100540 | 1.0098134 | -0.8788687 | -0.8670245 | -1.3067576 | 0.2976825 | 0.1493618 | 1.7864202 | -0.5689624 | med_low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.5773192 | 0.9671829 | -0.8494724 | -0.8670245 | -1.3067576 | 0.2976825 | 0.2487102 | 0.6899446 | -0.4058676 | med_low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.4250315 | 0.7042949 | -0.8558361 | -0.8670245 | -1.3067576 | 0.2976825 | 0.3104881 | 0.3020471 | -0.1231699 | med_low |
| -0.4872402 | 2.1155211 | -0.2723291 | 0.2270061 | -0.9559038 | 0.9600779 | -0.9677698 | -0.8670245 | -1.3067576 | 0.2976825 | 0.0286541 | 2.0454854 | -0.7429302 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.8420438 | 0.9742880 | -0.9530004 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3881485 | 0.6353309 | -0.6885653 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2083149 | 1.0737592 | -0.9415079 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.3832675 | -0.4928515 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.9217458 | 0.9281050 | -0.8620098 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.7963713 | -0.8951520 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2467426 | 1.0773117 | -0.7961887 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4202424 | -0.0074307 | -0.3623756 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.0588736 | 1.0346812 | -0.7237666 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | -0.0550427 | -0.3188837 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.1243431 | 1.0417863 | -0.6969823 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3185937 | -0.2146828 | 0.0507979 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.6584445 | 0.9529728 | -0.6293092 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3506875 | 0.3328548 | -0.4493595 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.7509558 | 1.0595490 | -0.6881492 | -0.6373311 | 0.1706618 | 1.2676838 | -1.0286891 | 0.6521351 | -0.7538032 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.0716829 | 1.0524439 | -0.7998930 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4161895 | 0.6031228 | -0.4819785 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.4876545 | 0.8854745 | -0.8681835 | -0.6373311 | 0.1706618 | 1.2676838 | 0.2363328 | 0.5947207 | -0.5580894 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.2410496 | 1.0595490 | -0.9237941 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4097270 | 0.2712393 | -0.5907084 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.6086307 | 1.0524439 | -1.0098459 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3873818 | 1.2136763 | -1.0038819 | med_low |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.1901952 | 1.0417863 | -1.0097984 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 0.8131756 | -0.5145975 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.1574604 | 0.8890270 | -1.0367727 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3440059 | 1.6113762 | -0.9277710 | med_high |
| -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -1.8013144 | 1.1163897 | -1.1186928 | -0.6373311 | 0.1706618 | 1.2676838 | 0.4406159 | 3.0467371 | -0.8842790 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -1.2547863 | 1.1163897 | -1.1746359 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 1.9838699 | -0.9930089 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.1622751 | 1.1163897 | -1.1318000 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 1.9278558 | -0.7538032 | high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.9664114 | 1.0382338 | -1.1630958 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4406159 | 2.3297568 | -1.1669767 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.2200834 | 1.1163897 | -1.1283332 | -0.5224844 | -0.0310742 | -1.7347012 | -2.0128627 | 2.1211044 | -0.9495170 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.9345550 | 1.1163897 | -1.0820306 | -0.5224844 | -0.0310742 | -1.7347012 | -2.0527336 | 0.5597119 | -0.7538032 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.9336767 | 0.9636304 | -1.1085299 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3837671 | 2.3633653 | -0.8625331 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.5636316 | 0.8961321 | -1.0758569 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0034610 | 2.1939227 | -0.5145975 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.9786758 | 0.9352101 | -1.0777090 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0528401 | 1.2318808 | -0.7755492 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.2314694 | 1.0204711 | -1.0338758 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1766361 | 0.2026221 | -0.1122969 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.2533631 | 1.1163897 | -1.0464131 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1651137 | 0.0877932 | -0.3188837 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -1.8112772 | 0.6900847 | -1.0375800 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1467118 | -0.0746476 | -0.7864221 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -0.8192718 | 1.0631016 | -1.0314063 | -0.5224844 | -0.0310742 | -1.7347012 | -1.0375614 | 0.4392816 | -0.3406296 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.2215067 | 0.9742880 | -0.9714740 | -0.5224844 | -0.0310742 | -1.7347012 | -0.3905371 | 0.3454580 | -0.6015814 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.1887719 | 0.4982475 | -0.9733261 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9428165 | 0.3314545 | -0.7538032 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.4412321 | 0.9032372 | -0.9776477 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9360253 | 0.4882939 | -1.0256279 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.9370190 | 1.0240236 | -0.9107344 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0740016 | -1.1291127 | 2.0405547 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.3111714 | 1.1163897 | -0.9677223 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0304949 | -0.8714479 | 0.1921468 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | 0.3207517 | 1.1163897 | -0.9636382 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0836407 | -0.7370141 | 0.0834169 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | -0.0492934 | 0.8535016 | -0.9482040 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1944691 | -1.0016807 | 0.4857174 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 1.7141136 | 0.7895559 | -0.8662839 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1944903 | -1.5296134 | 2.9865046 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | 2.1595909 | 1.0524439 | -0.8331359 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3607647 | -1.5030067 | 2.9865046 | med_high |
| -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | 2.9751133 | 0.8996847 | -0.7755306 | -0.5224844 | -0.0310742 | -1.7347012 | 0.3480587 | -1.3069574 | 2.9865046 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.6129005 | 0.8250813 | -0.6520568 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4210091 | -0.1418645 | 0.0181789 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.2613577 | 0.8677118 | -0.7178779 | -0.5224844 | -0.0310742 | -1.7347012 | -1.2762386 | -0.3981289 | 0.2682577 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 2.3403437 | 0.9813931 | -0.8306664 | -0.5224844 | -0.0310742 | -1.7347012 | 0.1382988 | -1.2537440 | 2.9865046 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5801657 | 0.3774611 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -1.4137053 | -0.0718469 | 0.1377818 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.0489109 | 0.9778406 | -0.8049744 | -0.5224844 | -0.0310742 | -1.7347012 | -0.6526548 | -0.2174835 | 0.1377818 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | 0.1670406 | 0.9458677 | -0.7278033 | -0.5224844 | -0.0310742 | -1.7347012 | -0.2917364 | -0.1866758 | -0.0253130 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5830122 | 0.9245525 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -0.7052317 | 0.2488337 | -0.5580894 | med_high |
| -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5758960 | 1.0204711 | -0.6678710 | -0.5224844 | -0.0310742 | -1.7347012 | -0.0935872 | -0.0872508 | -0.3732486 | med_high |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -1.0142570 | 0.7078474 | -0.5693769 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4406159 | 0.2852429 | 0.0616709 | med_low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.1869661 | 0.5515356 | -0.5455370 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4252810 | -0.5059560 | 0.1160358 | med_low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -0.6057842 | 0.0044442 | -0.5191326 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4004165 | -0.4219349 | 0.0073060 | med_low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.3719887 | -1.2602606 | -0.3147360 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3755520 | -1.0254866 | 0.7466691 | low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | -0.3766409 | -0.7593522 | -0.1140436 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4004165 | -0.3561184 | 0.0725439 | low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.0432179 | 0.1714136 | -0.2267846 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4263763 | -0.8910529 | 0.2247657 | low |
| -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.8188892 | 0.2069391 | -0.4177891 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3789476 | -0.8028307 | 0.8010341 | low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.9896793 | -0.3614676 | -0.4587729 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -1.0660969 | 1.5947622 | low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 2.1069307 | 0.5231153 | -0.5005640 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4259382 | -0.7132081 | 1.8774599 | low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.2001579 | -0.2264710 | -0.5685221 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.4485416 | 1.4860323 | low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 1.2387480 | 0.8392915 | -0.5197499 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4101651 | -1.0969046 | 1.6708731 | med_low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.3961839 | 0.9600779 | -0.4502247 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.9764743 | 1.0837317 | med_low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.9687130 | 0.7540305 | -0.3833114 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3759901 | 0.1858179 | 0.4204795 | med_low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.1873487 | 0.0079967 | -0.2447358 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3333809 | 0.0695886 | 0.7684151 | low |
| -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 2.2008652 | -0.5319896 | -0.2829652 | -0.7521778 | -1.2770905 | -0.3027945 | 0.3938444 | -1.1487176 | 2.9865046 | low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.7078757 | -0.9760573 | -0.0030596 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4074267 | -0.8364391 | 1.0293668 | low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.3862212 | -1.4023623 | 0.3664594 | -0.5224844 | -0.0607412 | -1.5037485 | 0.2866094 | -1.1333138 | 0.7901611 | med_low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 1.2814456 | -1.0542132 | 0.3664594 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4406159 | -1.0170845 | 1.3446835 | med_low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.9484050 | -1.6723554 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.2300893 | -1.0576947 | 1.5730162 | med_low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.6466760 | -1.3419691 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3618601 | -1.1151092 | 0.8662720 | low |
| 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 1.2714828 | -1.5018334 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3704038 | -1.3699732 | 1.5077783 | med_low |
| 2.0853880 | -1.1962619 | -0.2723291 | -1.3263561 | 0.7334942 | -2.0844502 | 1.1514203 | -0.9818712 | -0.8498849 | -1.3189864 | 0.4019500 | -1.0674972 | 0.9315099 | low |
| 2.0853880 | -1.1962619 | -0.2723291 | -1.3263561 | 0.4545372 | -1.7682741 | 1.1514203 | -0.9818712 | -0.8498849 | -1.3189864 | 0.2193548 | -1.1585201 | 0.7140502 | low |
| 2.9429307 | -1.5563017 | -0.2723291 | -1.1451305 | 2.2634882 | -1.2993386 | 0.8801579 | -0.6373311 | -0.9092190 | -1.8732728 | 0.4113700 | -1.3559697 | 2.9865046 | low |
| 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.4266171 | -1.2247352 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | 0.4406159 | -1.2005307 | 1.1707156 | low |
| 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.1704320 | -1.1359217 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | -0.0258945 | -0.5661712 | 0.8445260 | low |
| 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.4081148 | -1.0755284 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | 0.3891344 | -0.8448412 | 1.3120645 | low |
| 3.5860878 | -1.4090789 | -0.2723291 | -1.3090965 | 0.9825630 | -1.8926130 | 1.8323307 | -0.7521778 | -0.0370076 | -0.6723188 | 0.4406159 | -1.1333138 | 1.3446835 | low |
| 3.5860878 | -1.4090789 | -0.2723291 | -1.3090965 | 1.2102830 | -1.9423486 | 1.8323307 | -0.7521778 | -0.0370076 | -0.6723188 | 0.3026016 | -1.1487176 | 1.1272237 | low |
| 3.0501236 | -1.3274505 | -0.2723291 | -1.2055390 | -0.1745394 | -1.0719759 | 1.1753552 | -0.8670245 | -0.3574118 | -1.7347012 | 0.4063314 | -0.7314127 | 0.1704008 | low |
| 3.0501236 | -1.3274505 | -0.2723291 | -1.2055390 | 1.8863269 | -1.8784028 | 1.1753552 | -0.8670245 | -0.3574118 | -1.7347012 | 0.4239665 | -1.3363648 | 2.1492845 | low |
| 3.5860878 | -1.2327031 | -0.2723291 | -1.1960462 | 2.2321767 | -1.2567081 | 0.6282713 | -0.6373311 | -1.0931548 | -1.7347012 | 0.3954874 | -1.2383402 | 2.8234098 | low |
| 3.5860878 | -1.2327031 | -0.2723291 | -1.1960462 | 2.4897850 | -1.3028911 | 0.6282713 | -0.6373311 | -1.0931548 | -1.7347012 | 0.3710611 | -1.3685729 | 2.9865046 | low |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.5602402 | -1.6439351 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | -0.2496916 | 0.0073060 | med_low |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.0588736 | -0.5710675 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4183803 | -0.2356881 | 0.2030197 | med_low |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.7139512 | 0.1465458 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3587931 | 0.7571615 | -0.0035670 | med_low |
| -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.3140179 | -0.3365998 | 0.2109299 | -0.6373311 | -0.7786840 | 0.0667298 | 0.2699601 | 0.2810418 | 0.2030197 | med_low |
| -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.3387581 | 1.1163897 | 0.0379717 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | 1.4615386 | -0.2753917 | med_high |
| -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.4620360 | 0.8357389 | 0.0389689 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4006356 | 0.6465337 | -0.0905509 | med_low |
| -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.2533631 | 0.7114000 | -0.0617572 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4224331 | 1.5861699 | -0.3515026 | med_high |
| -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.6797932 | -0.5248845 | -0.0676459 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3753329 | 0.4728900 | -0.0144400 | med_low |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.1286129 | -1.2886809 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3191414 | -0.4583441 | 0.6053203 | med_low |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -1.2419771 | -2.0880028 | -0.0985619 | -0.6373311 | -0.7786840 | 0.0667298 | -0.0848244 | 2.3661660 | 0.1269088 | med_high |
| -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.1460744 | -0.9298742 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4047979 | -0.4457409 | 0.2682577 | med_low |
| -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | -0.5645100 | -0.4467286 | -0.3243289 | -0.5224844 | -0.7846174 | -0.9494620 | 0.3957065 | 0.1200013 | 0.0834169 | low |
| -0.4872402 | 0.4013236 | -0.2723291 | -0.0405174 | 0.5086207 | 0.5870610 | -0.1775851 | -0.5224844 | -0.7846174 | -0.9494620 | 0.3954874 | -0.4149332 | 0.6705582 | low |
| -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | -0.4748452 | 0.8961321 | -0.4301365 | -0.5224844 | -0.7846174 | -0.9494620 | 0.4406159 | 0.7375566 | -0.1122969 | med_low |
| -0.4872402 | 0.4013236 | 3.6647712 | -0.0405174 | 0.1257664 | 0.8463965 | -0.2050342 | -0.5224844 | -0.7846174 | -0.9494620 | 0.4060028 | -0.3015046 | 0.0507979 | med_low |
| -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.9484050 | 0.7078474 | -0.4432437 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3836576 | -0.4121325 | 0.4530985 | med_high |
| -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | -0.1716929 | 0.8073186 | -0.3547700 | -0.1779443 | -0.6006817 | -0.4875567 | 0.4224331 | 1.2332812 | -0.0905509 | med_high |
| -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.8459310 | 0.3241729 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3693085 | -0.3813247 | 0.5400824 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 0.4744627 | 0.4343017 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.4406159 | -0.7076067 | 0.8227800 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 2.8199790 | 0.3454882 | -0.4277145 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3108167 | -1.1921285 | 2.4211092 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 3.4732509 | 0.5124576 | -0.4277145 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2774085 | -1.1235113 | 2.9865046 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 2.4983245 | 0.6367966 | -0.2751294 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3363384 | -1.3335641 | 1.6382541 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.2501340 | 0.4023288 | -0.2751294 | -0.1779443 | -0.6006817 | -0.4875567 | 0.1687496 | -0.8812504 | 0.9858749 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.9944939 | -1.8322198 | -0.1994304 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2282272 | -1.2229363 | 2.6276960 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 0.3805282 | -1.6759080 | -0.1994304 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2592256 | -1.2453419 | 0.9750019 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | -0.4321477 | -0.0168711 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2374281 | -0.1404642 | 0.1921468 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.6045233 | 0.2957526 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2132208 | -1.0366895 | 0.9967478 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.9210298 | 0.1678611 | 0.0205904 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3202367 | -1.4259873 | 2.0840466 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.7929373 | 0.0648374 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2440002 | -1.2187352 | 2.8016638 | med_high |
| -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.6281737 | -0.0737117 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.0386218 | -0.6445909 | 0.7031772 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | -0.2827064 | -0.2513388 | -0.0679783 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2199025 | -0.2482913 | 0.1595278 | med_high |
| -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.4929649 | 0.2815424 | 0.1676191 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3480587 | -0.4359384 | 0.2791307 | med_high |
| -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 1.5276678 | 0.1074679 | 0.1676191 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3658034 | -1.1095078 | 0.9750019 | med_high |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.2794774 | -1.7789317 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2490389 | -0.8812504 | 0.1269088 | med_low |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.4573837 | -0.9369793 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2969057 | -0.7398148 | 0.0834169 | med_low |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.8715495 | -0.5071218 | 1.2067460 | -0.4076377 | -0.6422155 | -0.8570810 | 0.3787285 | -0.1782737 | -0.0579320 | med_low |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | -0.2698972 | -0.1234473 | 1.2067460 | -0.4076377 | -0.6422155 | -0.8570810 | 0.4156419 | -0.0354378 | -0.2645187 | med_low |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.1044176 | -0.5568574 | 1.5388905 | -0.4076377 | -0.6422155 | -0.8570810 | 0.1760884 | -0.2006793 | -0.0361860 | med_low |
| 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.1542314 | -2.1590536 | 1.5388905 | -0.4076377 | -0.6422155 | -0.8570810 | 0.1975573 | -1.0450916 | 0.1269088 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9843688 | 0.2815424 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1732405 | -0.0214342 | -0.5363434 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9672898 | 0.0577323 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3555071 | 0.8131756 | -0.4384865 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.2513949 | -1.1963149 | 2.0232877 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3670082 | -0.4891518 | 0.1921468 | med_high |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.0834514 | 0.3774611 | 2.0232877 | -0.2927910 | -0.4642132 | 0.2976825 | 0.2132208 | -0.3505170 | -0.2210268 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2111614 | -0.6918540 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1975573 | -0.4387391 | 0.2138927 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.6167877 | -1.8144571 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4060028 | -0.8532433 | 0.3987335 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2880169 | -1.9743215 | 1.7104241 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4338247 | -0.9456666 | 0.2030197 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2182776 | -2.1199757 | 1.7104241 | -0.2927910 | -0.4642132 | 0.2976825 | 0.2234076 | -1.2691479 | 0.2465117 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.9569445 | -2.1945790 | 2.4275218 | -0.2927910 | -0.4642132 | 0.2976825 | 0.3222084 | -1.2775500 | 0.7684151 | med_low |
| 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 2.8100163 | -2.1377384 | 2.4275218 | -0.2927910 | -0.4642132 | 0.2976825 | 0.4406159 | -1.2761497 | 2.2036495 | med_high |
| 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.2513949 | -1.2993386 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.3966923 | -0.8518430 | -0.0688050 | low |
| 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.5815890 | -1.7576164 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.4217758 | -0.4765487 | -0.1775348 | low |
| 3.3717021 | -1.0767345 | -0.2723291 | -1.3867646 | 1.6642999 | -1.2211827 | 1.2067460 | -0.7521778 | -0.9744866 | -1.1804147 | 0.3249467 | -1.3363648 | 2.3341253 | low |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 3.4433626 | 0.6510068 | -0.9469692 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3617506 | -1.0548940 | 2.9865046 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.4920866 | 1.1163897 | -0.9025187 | -0.5224844 | -0.8558183 | -2.5199404 | 0.2915385 | -0.6810000 | 1.4642863 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 0.7932707 | 1.1163897 | -0.8473829 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3861769 | -0.8056314 | 0.8227800 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.3070641 | 0.4698271 | -0.7992281 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3957065 | -0.4289367 | 1.2250806 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.7582344 | 0.7398203 | -0.7860734 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3471824 | -0.7552187 | 2.2362684 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 3.0078481 | 0.8144237 | -0.7154559 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3306426 | -0.9442662 | 2.8560287 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.4835471 | 0.9209999 | -0.8150422 | -0.5224844 | -0.8558183 | -2.5199404 | 0.4024977 | -0.1964782 | 0.9206369 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.3113338 | 0.8179762 | -0.8856597 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3419247 | -0.6375891 | 1.5186513 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | -1.0313360 | -0.2051558 | -0.8588754 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3913251 | -0.3085064 | 0.0290519 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.0380698 | 0.5692983 | -0.7893502 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3000823 | 0.2992464 | 0.8880180 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 2.8640998 | -0.0559490 | -0.6522468 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3052304 | -0.7300124 | 2.9865046 | med_high |
| 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 1.6870719 | -0.5675150 | -0.4383522 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3683227 | -1.3293630 | 2.2797604 | med_high |
| 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | -0.5189660 | -0.2513388 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3797143 | 0.1396062 | -0.1992808 | med_low |
| 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.6100540 | -0.9405319 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3502494 | 0.0485834 | -0.1557889 | med_high |
| 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.0635259 | -1.8570876 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.4406159 | -0.8490423 | 0.2900036 | med_low |
| 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | 0.3606027 | -0.3508100 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.4193661 | -0.6894022 | 0.2030197 | med_low |
| 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | 2.0016102 | -0.5959353 | 0.2713846 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3734708 | -0.8504426 | 1.3773024 | med_low |
| 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 0.6737177 | -1.2673657 | 0.1341862 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -1.2775500 | 1.0728588 | low |
| 1.2278453 | -0.6889993 | -0.2723291 | -0.9293857 | 0.8103497 | -0.9156641 | 0.2242746 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -1.3545694 | 1.0293668 | med_low |
| 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 1.3981521 | -0.6954065 | 0.4711747 | -0.6373311 | -0.9151524 | -0.3951756 | 0.3568215 | -0.9246613 | 1.1598427 | med_low |
| 1.2278453 | -0.6889993 | 3.6647712 | -0.9293857 | 0.7704987 | -1.4556504 | 0.5070771 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4028263 | -1.1893278 | 1.1489697 | low |
| 1.2278453 | -0.6889993 | -0.2723291 | -0.9293857 | 0.2809007 | -1.2957860 | 0.1639624 | -0.6373311 | -0.9151524 | -0.3951756 | 0.4406159 | -0.7650212 | 0.7140502 | low |
| 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 0.7505732 | -1.2922335 | 0.1451564 | -0.5224844 | -1.1406221 | -1.6423201 | 0.4406159 | -1.0927035 | 1.3664294 | med_low |
| 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 2.1852094 | -0.1447626 | 0.4272465 | -0.5224844 | -1.1406221 | -1.6423201 | 0.3355717 | -1.2453419 | 2.4863472 | low |
| 0.3703025 | -1.1379558 | -0.2723291 | -0.9647679 | 0.9726003 | -1.1146064 | 0.6884411 | -0.5224844 | -1.1406221 | -1.6423201 | 0.3894630 | -1.1291127 | 1.3990484 | low |
| 0.3703025 | -1.1379558 | 3.6647712 | -0.9647679 | 1.9361406 | -0.6705387 | 0.6728644 | -0.5224844 | -1.1406221 | -1.6423201 | 0.2234076 | -1.3503683 | 2.5515851 | low |
| 3.3717021 | -1.4469778 | 3.6647712 | -1.3263561 | 2.3318042 | -1.5551216 | 0.9925190 | -0.9818712 | -1.2474235 | -2.2427971 | 0.4255000 | -1.3293630 | 2.9865046 | low |
| 3.3717021 | -1.1904313 | -0.2723291 | -1.3349859 | 1.1433903 | -1.6972232 | 1.6679681 | -0.9818712 | -0.7312167 | -1.4575580 | 0.4167372 | -0.6725979 | 1.0511128 | low |
| 1.8710023 | -1.2953821 | -0.2723291 | -1.4299136 | 0.2396264 | -1.3028911 | 1.6679681 | -0.9818712 | -0.6422155 | -1.4575580 | 0.4167372 | -0.6193846 | -0.0579320 | low |
| 2.9429307 | -1.3668070 | -0.2723291 | -1.4644327 | -0.0777584 | -1.3171013 | 2.5141909 | -0.9818712 | -0.9922868 | -0.1180323 | -0.1651137 | 0.0387809 | -0.2645187 | low |
| 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | -0.1076467 | -1.3242064 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.4406159 | -0.7720229 | 0.0725439 | low |
| 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | 0.0432179 | -0.8161929 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.4406159 | -0.7076067 | -0.0253130 | low |
| 1.7638095 | -0.8478833 | -0.2723291 | -1.2918369 | 0.3990304 | -1.6226198 | 1.6726696 | -0.4076377 | -0.6837494 | -0.8570810 | 0.1648063 | -0.4401395 | 0.2465117 | low |
| 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 0.8203125 | -1.4449928 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.3055571 | 0.6488122 | low |
| 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 1.2287853 | -1.4520979 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.2733490 | 1.6056352 | low |
| 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 0.4915417 | -1.6048571 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.1137088 | 0.5835743 | low |
| -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.2243532 | -1.7824842 | 0.8109650 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.5703722 | 0.1486548 | med_low |
| -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.3922967 | -0.9334268 | 0.8109650 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.3155082 | -0.0905509 | low |
| -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | 0.5598577 | -1.3313114 | 1.0283264 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | -0.8938536 | 0.6596852 | med_low |
| -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | 0.3762584 | -0.6243557 | 1.0283264 | -0.6373311 | -0.7074831 | -1.1342242 | 0.3962541 | -0.7370141 | 0.4965904 | low |
| -0.4872402 | 0.4056965 | -0.2723291 | -1.0156836 | -0.7039885 | -0.3756778 | 1.1991001 | -0.6373311 | -0.7074831 | -1.1342242 | 0.4406159 | 0.4462833 | -0.2427728 | med_low |
| 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 0.0859154 | -1.7220910 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.1266881 | -1.0758993 | -0.0035670 | low |
| 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 1.0764975 | -2.0808977 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.1632728 | -1.1081074 | 0.7031772 | low |
| 2.5141594 | -1.2968398 | -0.2723291 | -1.3349859 | 0.8345450 | -0.7522472 | 1.9151531 | -0.5224844 | -0.2980777 | -1.6885107 | 0.3744566 | -0.9218606 | 0.2465117 | low |
| 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.4346117 | -1.0009251 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.4280193 | -0.4415399 | -0.0579320 | low |
| 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.2994029 | -1.7824842 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.2950436 | -0.5577691 | 0.4204795 | med_low |
| 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.9925258 | -1.8073520 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.3697466 | -1.0913032 | 1.1489697 | med_low |
| 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 1.3540313 | -0.9760573 | 0.1077818 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4053456 | -0.8014303 | 1.4751593 | low |
| 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 0.4716162 | -0.3721252 | -0.2018524 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4018404 | -0.5213599 | 0.6379392 | low |
| 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 1.6159094 | 0.1181255 | -0.3304551 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4406159 | -0.8658465 | 1.1815886 | low |
| 0.9277053 | -1.3055857 | -0.2723291 | -0.7136410 | 0.8032335 | 0.0612849 | -0.2908010 | -0.2927910 | -1.1050216 | -0.0256513 | 0.4406159 | -0.7174092 | 0.6161933 | low |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4986579 | 0.4946949 | -0.2267846 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -1.1361145 | 0.0290519 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.4449570 | 0.2886475 | -0.3288880 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4333866 | -0.3757233 | -0.2427728 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -1.8667839 | -1.0932912 | -0.6058017 | -0.6373311 | -0.6184819 | -0.0256513 | -0.0681750 | -0.0018293 | -0.6994382 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.2314694 | -0.5604099 | -0.5483863 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -0.9344638 | -0.0470590 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.3723712 | 0.7753457 | -0.4563984 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4340438 | -0.1306617 | -0.3406296 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.0265214 | 0.5053525 | -0.2527616 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4021690 | -0.6655962 | -0.1014239 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4018769 | 0.6652169 | -0.0915333 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4273621 | -0.4723476 | 0.1377818 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.8249648 | 0.3241729 | 0.0712146 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4353582 | -0.1614694 | -0.6885653 | med_low |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.5275055 | 0.5195627 | 0.0966692 | -0.6373311 | -0.6184819 | -0.0256513 | 0.3727041 | 0.7949710 | -0.5145975 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.7153745 | 0.1110204 | 0.1123884 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | 0.4602869 | -0.2971377 | med_low |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.1385756 | -0.0488439 | -0.1246813 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4221044 | -0.3211096 | 0.0616709 | med_high |
| -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | -0.2442787 | -0.3472574 | 0.0982364 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4332770 | 0.0107739 | -0.1666618 | med_high |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.2011986 | -0.5781726 | 0.3539696 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.7636208 | 0.1377818 | med_low |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.1300361 | -0.5071218 | 0.3539696 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.8098324 | 0.0616709 | med_low |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.3467527 | -0.6634336 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.6936032 | -0.2318998 | med_high |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.8206950 | 0.2033865 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.3774141 | -0.1278610 | -0.4384865 | med_high |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.1855429 | -1.0115827 | 0.4397839 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.9148588 | 0.2682577 | med_high |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.2083149 | -1.9139283 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4053456 | -1.0604955 | 0.2247657 | med_low |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | 0.0389481 | -1.4094674 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | -0.9106578 | 0.0507979 | med_high |
| -0.4872402 | -0.5476072 | -0.2723291 | -0.5324154 | -0.2869762 | -0.8836912 | 0.7697438 | -0.5224844 | -0.7193499 | 0.5286352 | 0.4406159 | 0.0191760 | -0.0361860 | med_low |
| -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.5929750 | -1.5195961 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2822280 | -0.3757233 | -0.3515026 | low |
| -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | 0.0688364 | -1.8251147 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2030341 | -0.7440159 | 0.0073060 | low |
| -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.2001579 | -1.2922335 | 0.9871052 | -0.6373311 | 0.1291279 | -0.7185093 | 0.1303027 | -0.4989543 | -0.2971377 | low |
| 1.0134596 | -0.7400171 | -0.2723291 | -1.0079168 | -0.8235415 | -1.4272301 | 1.3514003 | -0.9818712 | -0.6184819 | -0.7185093 | 0.4090698 | -0.0312367 | -0.5907084 | low |
| 1.0134596 | -0.7400171 | -0.2723291 | -1.0079168 | -0.3609852 | -1.6084097 | 1.3514003 | -0.9818712 | -0.6184819 | -0.7185093 | 0.0610765 | -0.6753986 | -0.3406296 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | 0.0446411 | -1.0826335 | 1.2648262 | -0.5224844 | -1.0931548 | 0.8057784 | 0.3618601 | -0.9764743 | -0.0361860 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | 0.0361016 | -1.0684234 | 1.2648262 | -0.5224844 | -1.0931548 | 0.8057784 | 0.3584645 | -0.8266367 | -0.1992808 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.3524457 | -1.2105250 | 1.0401514 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.6501923 | -0.1557889 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5915517 | -0.7913251 | 0.6819824 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.3995293 | -0.3297567 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5545472 | -0.3188371 | 0.8642962 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4177230 | -0.2931025 | -0.4384865 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.3211342 | -1.1110539 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4322912 | -0.5801747 | -0.2101538 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4264547 | -0.8232980 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4079314 | -0.3841216 | low |
| -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4506500 | -0.3579151 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4709472 | -0.4167406 | low |
| 1.0134596 | -1.4017907 | -0.2723291 | -0.9725347 | 1.3611476 | -0.6847489 | 1.5400303 | -0.9818712 | -0.7371501 | -1.3651769 | 0.4169563 | -1.0030810 | 1.1054777 | low |
| -0.4872402 | -1.3478576 | -0.2723291 | -0.3166707 | 0.3634492 | -0.3152846 | 1.1738830 | -0.9818712 | 0.0816606 | -1.1804147 | 0.3645985 | -0.5605698 | -0.6559463 | low |
| 1.8710023 | -1.0723615 | -0.2723291 | -0.6100835 | 0.5854762 | -0.4325184 | 0.9199069 | -0.5224844 | -0.2268768 | -0.3951756 | 0.4406159 | -0.7664215 | 0.1486548 | low |
| 1.8710023 | -1.0723615 | -0.2723291 | -0.6100835 | 0.8388147 | -1.4378877 | 1.2681505 | -0.5224844 | -0.2268768 | -0.3951756 | 0.3428010 | -1.1263120 | 0.9423829 | low |
| -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.3851804 | -0.7131692 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.3172793 | -0.2973036 | -0.5472164 | low |
| -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.5502775 | -0.5781726 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.0869268 | 0.0023717 | -0.5798354 | low |
| 3.1573164 | -1.0184285 | -0.2723291 | -1.0847220 | 0.3292912 | -1.4520979 | 2.2511443 | -0.6373311 | -0.3396116 | -0.2566040 | 0.3916537 | -0.8812504 | 0.0616709 | low |
| 2.9429307 | -1.3303658 | -0.2723291 | -1.0329432 | 0.4986579 | -1.3810470 | 2.1602961 | -0.6373311 | -0.7608838 | -0.6723188 | 0.3753329 | -0.9330634 | 0.2138927 | low |
| 1.2278453 | -1.4411472 | -0.2723291 | -1.0847220 | 0.9313260 | -1.2105250 | 2.3730984 | -0.9818712 | -0.4345461 | 0.5748257 | 0.3633936 | -0.9470669 | 0.4422255 | low |
| 1.2278453 | -1.4411472 | -0.2723291 | -1.0847220 | 0.2922867 | -0.8588234 | 2.3730984 | -0.9818712 | -0.4345461 | 0.5748257 | 0.4406159 | -0.9344638 | 0.0399249 | low |
| 2.0853880 | -1.3770106 | -0.2723291 | -1.2400582 | 0.4189559 | -1.1607894 | 3.2840500 | -0.6373311 | 0.0163931 | -0.0718418 | 0.1545100 | -1.0030810 | 0.1704008 | low |
| 2.0853880 | -1.3770106 | -0.2723291 | -1.2400582 | -0.5702030 | -1.7789317 | 3.2840500 | -0.6373311 | 0.0163931 | -0.0718418 | 0.3905583 | -0.6810000 | -0.4276135 | low |
| 3.3717021 | -1.3289081 | -0.2723291 | -1.2486880 | 0.6310202 | -1.1536844 | 3.9566022 | -0.5224844 | -1.3126910 | -0.6723188 | 0.3043541 | -1.1417159 | 0.8227800 | low |
| 2.9429307 | -1.3449423 | -0.2723291 | -1.2227986 | -0.8847413 | -1.6581453 | 3.2248775 | -0.6373311 | -0.4404796 | 1.6372081 | 0.2861713 | -0.6445909 | -0.4711055 | low |
| 2.9429307 | -1.3449423 | -0.2723291 | -1.2227986 | -0.4961940 | -1.7434063 | 3.2248775 | -0.6373311 | -0.4404796 | 1.6372081 | 0.2121255 | -0.9918782 | -0.2101538 | med_low |
| -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.1033769 | 1.0240236 | -0.7944316 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2306369 | 0.6927453 | -0.5145975 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | 0.1570779 | 0.7966610 | -0.6125452 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3797143 | 0.0863929 | -0.0905509 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.2243532 | 0.5266678 | -0.5092547 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4245142 | -0.1642701 | 0.0181789 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.2457019 | 0.4520644 | -0.6106931 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3731422 | 0.0023717 | 0.0073060 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | 0.1613476 | 0.6900847 | -0.6063715 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1959143 | -0.6810000 | 0.2682577 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.0478701 | 0.8002135 | -0.7121316 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0659843 | 0.2152253 | -0.2862647 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -1.3131396 | 0.9813931 | -0.8032647 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2641547 | -0.3449156 | -0.1884078 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.6854862 | 0.7256101 | -0.8977222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0398054 | 0.2782411 | -0.6233273 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 1.4092873 | 3.5515296 | 0.5089051 | -0.8977222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0232656 | -1.0310881 | -0.0688050 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -3.8764132 | 0.6865322 | -1.0361553 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0216226 | -0.7748236 | 0.5400824 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -1.8810164 | 0.8108711 | -0.9700968 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4451952 | 0.1886186 | -0.0688050 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -3.4465917 | 1.1163897 | -1.0848799 | 1.6596029 | 1.5294129 | 0.8057784 | -2.4673242 | 0.0947950 | 0.0616709 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -1.8710537 | 1.1163897 | -1.1694595 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2064297 | -1.3153595 | 2.9865046 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 0.6584956 | 0.5669739 | 1.0027084 | -1.1579669 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2043485 | -1.2495430 | 2.9865046 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 0.6584956 | 1.0409163 | 1.0275762 | -1.2312439 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3874913 | -1.3573701 | 2.9865046 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.6584956 | -0.0976839 | 1.1163897 | -1.2470580 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1037952 | -0.4373388 | 2.9865046 | high |
| -0.4872402 | 1.0149946 | 3.6647712 | 0.9777978 | -0.5830122 | 0.7469254 | -1.2658165 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0963256 | -0.5283617 | 2.9865046 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -1.9621417 | 1.1163897 | -1.2446360 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.0971497 | -0.9495170 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -3.0551979 | 1.1163897 | -1.2623023 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.5452624 | -0.9495170 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 1.4636216 | 1.0417863 | -1.1771529 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.1101988 | -0.8190411 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.5185834 | 0.8783694 | -1.1635707 | 1.6596029 | 1.5294129 | 0.8057784 | 0.0695107 | 1.4825438 | -0.9386440 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.7249547 | 1.0737592 | -1.1573496 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.2024734 | -1.0038819 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.1357291 | 0.9813931 | -1.1440049 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.5455597 | -1.0256279 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | -0.0877212 | 1.1163897 | -1.1440049 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4060028 | 1.2780924 | -1.3409445 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.9726003 | 0.8286338 | -1.1295680 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.6381316 | -1.3191985 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.3705654 | 1.0844168 | -1.0807958 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.1800678 | -1.2648336 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.0654941 | 1.1163897 | -1.0517320 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.5329565 | -1.2213417 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.0882661 | 1.1163897 | -1.0741947 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.6673903 | -1.1126118 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -2.7278503 | 0.8037660 | -1.1186453 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7759914 | 2.5174040 | -1.4931663 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.4341159 | 1.0488914 | -1.1250089 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.5426103 | -1.6671342 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -2.3236472 | 1.1163897 | -1.1054906 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.1883213 | -1.3083256 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.8283562 | 0.7433728 | -1.0811757 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.7078519 | -1.6453882 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.9991462 | 1.1163897 | -1.0474104 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1779505 | 2.5160036 | -1.3409445 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.2732886 | 1.0773117 | -0.9815894 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.1478597 | -1.1995957 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -0.8135788 | 1.0098134 | -0.8873694 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4135607 | 0.6241280 | -0.8081681 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -0.3325202 | 0.4946949 | -0.7727762 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2377567 | 0.8551861 | 0.0725439 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.7771192 | 1.0098134 | -0.9616911 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.8242297 | -1.3953095 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.1304187 | 0.8535016 | -0.9516232 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.3524598 | -0.9495170 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.5659332 | 0.9281050 | -0.9559448 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5177013 | -1.0691198 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.2652449 | 1.0737592 | -0.9827291 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3867246 | 0.6255284 | -1.0256279 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1713104 | 0.9742880 | -1.0059517 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.9406076 | -1.0908658 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.7651883 | 1.0773117 | -1.0265623 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3989925 | 1.0176270 | -1.5257853 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.1836238 | 1.1163897 | -1.0948528 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 2.5118026 | -1.9063399 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.6157470 | 0.3277255 | -1.0897239 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2027938 | 2.4249808 | -1.7649910 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.4236082 | 1.1163897 | -1.0477428 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.9768681 | -1.8411020 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.0830689 | 1.1163897 | -1.0547238 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.0736410 | -1.6671342 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1698871 | 1.1163897 | -1.0239029 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2128922 | 1.0722407 | -1.1343578 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.3316418 | 0.9742880 | -0.9936043 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.9966217 | -1.5475313 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -1.0726103 | 0.5977186 | -1.0389097 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2980894 | 2.0622896 | -1.5257853 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -0.8562763 | 1.1163897 | -1.1253414 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3099404 | 1.4461347 | -1.9063399 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.9001297 | -3.0551979 | 1.1163897 | -1.2427839 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1483760 | 1.4965474 | -1.1561037 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.9001297 | -0.9630200 | 1.1163897 | -1.1919222 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2692816 | -0.0732473 | 0.5835743 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -0.9502108 | 1.0417863 | -1.1114268 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4604205 | 1.9250551 | -0.5798354 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | 0.8075032 | 1.1163897 | -1.1062979 | 1.6596029 | 1.5294129 | 0.8057784 | -1.9422126 | 0.9980220 | 0.5400824 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -0.7509558 | 1.1163897 | -1.1312301 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8783565 | -0.3561184 | -0.8190411 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | 0.5299694 | 1.1163897 | -1.0768541 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5229148 | 1.1996727 | -0.5798354 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -2.3578052 | 1.1163897 | -1.0643168 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5914839 | 3.0411357 | -0.5037245 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -1.6077524 | 1.1163897 | -1.0474579 | 1.6596029 | 1.5294129 | 0.8057784 | -1.5959718 | 1.0400326 | -0.6776923 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | -2.5129395 | 1.1163897 | -1.0147848 | 1.6596029 | 1.5294129 | 0.8057784 | -2.9399686 | 3.4066275 | -1.6888801 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.2125846 | 1.1163897 | -0.9309651 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6083523 | 2.2961484 | -1.6671342 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.7078757 | 0.7895559 | -0.9381836 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6705683 | 1.8396336 | -1.6345152 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -1.3956881 | 0.7291627 | -1.0198662 | 1.6596029 | 1.5294129 | 0.8057784 | -2.5117955 | 1.9586635 | -1.3191985 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.4663057 | 1.1163897 | -0.9462094 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7266503 | 1.1156516 | -1.4931663 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | 0.7676522 | 0.2815424 | -0.9502935 | 1.6596029 | 1.5294129 | 0.8057784 | -3.3761377 | 1.4125262 | -1.5366583 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | 0.1798499 | 1.1163897 | -0.9194726 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4154016 | 0.3314545 | -0.6342003 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.4092873 | -0.3965665 | 0.9494202 | -0.9120166 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4019288 | 0.4266784 | -0.9060250 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.9060900 | 0.6758745 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7133373 | 0.2026221 | -0.1884078 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.2585112 | 0.5870610 | -0.8421115 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8792328 | 1.4895456 | -0.9930089 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -1.0242198 | 0.0719425 | -0.8223082 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8668553 | 0.6311298 | -1.1778497 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.5531240 | 0.9529728 | -0.8953952 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8227126 | 1.6435843 | -1.5475313 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.6370957 | -0.3152846 | -0.8536040 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6368314 | 0.4252781 | -1.3409445 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.1176094 | 0.3596983 | -0.9175730 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7006904 | 0.2614369 | -1.2648336 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.1304187 | 0.3383831 | -0.8830478 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8473018 | 1.2416833 | -1.2539606 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | 0.1357291 | 0.9600779 | -0.8675661 | 1.6596029 | 1.5294129 | 0.8057784 | -3.2417380 | 1.6001734 | -1.4170554 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.0901851 | 0.6225864 | -0.8274371 | 1.6596029 | 1.5294129 | 0.8057784 | -2.9927645 | 0.6983467 | -0.8734060 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.7804615 | 0.9138948 | -0.8105782 | 1.6596029 | 1.5294129 | 0.8057784 | -3.0159860 | 0.9854189 | -0.9168980 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | 0.1997754 | 0.2211492 | -0.7572945 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8339385 | -0.0872508 | -0.6994382 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.2154311 | 0.6865322 | -0.7024911 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8094026 | 0.4994967 | -0.8951520 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1090699 | 0.9387626 | -0.7469417 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8045831 | 0.3524598 | -1.1778497 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.4901184 | 0.9245525 | -0.7932444 | 1.6596029 | 1.5294129 | 0.8057784 | -2.7035916 | 1.4867449 | -0.9930089 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2510124 | 0.8783694 | -0.8512296 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6057234 | 0.7557611 | -1.4061824 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.1887719 | 1.1163897 | -0.8932106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8047489 | 1.9320568 | -1.5040393 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.4976172 | 0.6865322 | -0.9376612 | 1.6596029 | 1.5294129 | 0.8057784 | -3.1515905 | 2.9921233 | -1.5366583 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.9359783 | 0.8996847 | -0.9392759 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.4321312 | -1.0582468 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.6641375 | 0.8463965 | -0.9160058 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3809192 | 1.3243041 | -1.3083256 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.1727336 | 1.0169185 | -0.8215484 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3207844 | 0.9616129 | -0.5907084 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.0934142 | 1.1163897 | -0.8501848 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4273621 | 0.5513097 | -0.4493595 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2851704 | 1.1163897 | -0.8627221 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3292186 | 0.8677893 | -0.7755492 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.6129005 | 0.9956033 | -0.9020438 | 1.6596029 | 1.5294129 | 0.8057784 | -1.2722954 | 1.5595632 | -1.2757066 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2481659 | 0.9316575 | -0.8582106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.4351771 | 1.5861699 | -1.1669767 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.0802224 | 0.9884982 | -0.8182715 | 1.6596029 | 1.5294129 | 0.8057784 | -0.4235072 | 0.7193520 | -0.8299141 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | -0.0478701 | 0.9956033 | -0.7584343 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3488254 | 0.5303045 | -1.0799928 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1418047 | 1.0702067 | -0.7282307 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.7669640 | -0.9168980 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.1883894 | 1.0559965 | -0.7646079 | 1.6596029 | 1.5294129 | 0.8057784 | -0.5746657 | 0.9322055 | -1.0365009 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.6609085 | 0.8535016 | -0.6987869 | 1.6596029 | 1.5294129 | 0.8057784 | -3.9033305 | 0.6703397 | -0.9930089 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5271229 | 1.0524439 | -0.6837801 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0151600 | 0.7109499 | -0.7972951 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0175994 | 0.8250813 | -0.6776064 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3112548 | 0.6465337 | -0.6994382 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 1.5774816 | 1.0915219 | -0.6374774 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2102634 | 0.5723150 | -0.5145975 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.6310202 | 0.9067897 | -0.6168668 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8336662 | 0.8481844 | -0.8299141 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.3421004 | 0.6367966 | -0.6455032 | 1.6596029 | 1.5294129 | 0.8057784 | -3.3490825 | 0.7669640 | -0.9168980 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4392640 | 0.6865322 | -0.5767378 | 1.6596029 | 1.5294129 | 0.8057784 | -3.7920428 | 0.8901949 | -1.0691198 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4961940 | 0.4165390 | -0.4824229 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8684983 | 0.6003221 | -0.9821359 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0232924 | 0.5373254 | -0.4805707 | 1.6596029 | 1.5294129 | 0.8057784 | -0.9251783 | 0.5008971 | -0.8299141 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.2898227 | 0.5621932 | -0.5117241 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.2866432 | -0.2753917 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5925924 | 0.7611355 | -0.5687120 | 1.6596029 | 1.5294129 | 0.8057784 | -1.1111691 | 0.5275038 | -0.6668193 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.1300361 | 0.7042949 | -0.5831490 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3807001 | 0.2796414 | -0.5254704 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0460644 | 0.5124576 | -0.5036983 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.1872182 | -0.3297567 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.3250214 | 0.7575830 | -0.4717851 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4068791 | -0.3309120 | -0.2536457 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.1076467 | -0.1127897 | -0.3949464 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.0793911 | -0.1231699 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.7481093 | -0.7238268 | -0.3459843 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2439790 | 0.2068231 | -0.2862647 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.8656106 | -0.4734220 | 0.5728508 | -0.4385897 | 1.6596029 | 1.5294129 | 0.8057784 | -3.6657487 | 0.6297295 | -0.3841216 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.4008362 | 0.9209999 | -0.5958763 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2780445 | 1.2136763 | -0.3732486 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.5104265 | 0.0861526 | -0.4210659 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1321648 | 0.7669640 | -0.3732486 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.8135788 | -0.4218608 | -0.4612898 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.2950453 | -0.2645187 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.1674232 | 0.5479830 | -0.3617035 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5092992 | -0.2862647 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.0791817 | 0.7860033 | -0.3304076 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4234189 | 0.0303788 | -0.3188837 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | 0.2168544 | 0.2282543 | -0.4267172 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4019500 | 0.2390312 | 0.0725439 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | 0.9896793 | -0.0346338 | -0.5993905 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1972287 | -0.1390638 | 0.7901611 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -1.2206283 | 0.9529728 | -0.6483526 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0448441 | 0.7683643 | -0.9495170 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.1745394 | 1.0240236 | -0.7546351 | 1.6596029 | 1.5294129 | 0.8057784 | -0.5905484 | 1.6029741 | -1.0038819 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | 0.2837472 | 0.8890270 | -0.7074776 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4330580 | 0.8439833 | -0.6342003 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -1.3956881 | 1.0204711 | -0.8046419 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0788000 | 1.7164026 | -1.1452307 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.1418047 | 0.9991558 | -0.7714940 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2522154 | 0.7529604 | -0.8625331 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.0791817 | 0.6900847 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2918671 | 0.0639872 | -0.1231699 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.0606794 | -0.1376575 | -0.1761129 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | -0.2678962 | 0.0507979 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | 0.6623317 | 0.2247018 | -0.2200411 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3986639 | -0.6880018 | 0.1269088 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | 1.1049625 | 0.2993051 | -0.1825715 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4228712 | -0.7902275 | 0.2682577 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | -0.1958536 | -0.7438395 | -1.0044776 | 0.1440166 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3970209 | -0.3127075 | -0.0796779 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5887052 | -0.9476370 | -0.0337381 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1539623 | 0.0961953 | -0.2101538 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | 0.0389481 | -0.5923828 | 0.0933924 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3499208 | -0.2903018 | -0.1449159 | med_high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.2428554 | 0.3987763 | -0.1183177 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3943920 | 0.3258531 | -0.3732486 | high |
| -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5403147 | -0.5461998 | -0.3052380 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3455394 | -0.1684712 | -0.2101538 | high |
| -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.1822006 | 0.8570542 | -0.9375187 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4207900 | 0.7571615 | -0.7972951 | med_low |
| -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.2391306 | 1.0559965 | -0.9686246 | -0.6373311 | 1.7964164 | 0.7595879 | -0.1382776 | 1.5847695 | -1.6888801 | med_low |
| -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.6959939 | 1.0453389 | -0.9367114 | -0.6373311 | 1.7964164 | 0.7595879 | -0.4189067 | 2.3843705 | -1.5692773 | med_low |
| -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -0.4293012 | 1.0737592 | -0.9151035 | -0.6373311 | 1.7964164 | 0.7595879 | 0.3662415 | 0.7585618 | -0.9712629 | med_low |
| -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -0.4293012 | 0.5302203 | -0.8002729 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4406159 | 0.0975957 | -0.2645187 | med_low |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.8221183 | -0.5177794 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | -0.0900515 | -0.0796779 | med_low |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.5104265 | -0.9227692 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.1312041 | 0.2138927 | med_high |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.8747785 | -1.4130199 | -0.4732098 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4010737 | 0.6927453 | 0.0616709 | med_low |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -1.2732886 | 0.1536509 | -0.4732098 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 1.1884699 | -0.3080107 | med_high |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.6982955 | 0.0719425 | -0.4285218 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.2026221 | -0.4602325 | med_high |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.3780642 | -0.1163422 | -0.6581830 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.0373805 | -0.1449159 | med_low |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -1.0185268 | 0.1749662 | -0.6625521 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4282384 | 0.3426573 | -0.5472164 | med_low |
| -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.3666782 | 0.3952238 | -0.6158695 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.2348302 | -0.6233273 | med_low |
| -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.4388814 | 0.0186544 | -0.6251775 | -0.9818712 | -0.8024176 | 1.1753027 | 0.3868341 | -0.4177339 | -0.0144400 | low |
| -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | -0.2343159 | 0.2886475 | -0.7159308 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.5003546 | -0.2101538 | low |
| -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.9839863 | 0.7966610 | -0.7729187 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.9820757 | 0.1486548 | low |
| -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.7249547 | 0.7362677 | -0.6677760 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4028263 | -0.8644462 | -0.0579320 | med_low |
| -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | -0.3624084 | 0.4343017 | -0.6126402 | -0.9818712 | -0.8024176 | 1.1753027 | 0.4406159 | -0.6683969 | -1.1561037 | low |
We checked the scaled crime rate values and defined the quantiles of the crime data, which where then named as low (0-25%), medium low (25-50%), medium high(50-75%) and high (75-100%).Then the crime rate quantiles definintions were added instead of the crim values as “crime” variable into the ‘boston_scaled’ data set.
Quantiles:
0% - 25% -50% - 75% -100% of the data distribution –> quantiles give the values how many values are in which part of the data distribution. The median is the center of the quantiles (were the upper and lower 50 % quantiles meet).
1.3.3. Define training and test data set
We are going to use 80% of the observations (the rows) to train the linear discriminant analysis model.
Therefore we prepare a training data set ‘boston_train’ including 80% of the rows and including the “crime” variable.
The test data set ‘boston_test’ will include 20% of the rows and we remove the “crime” variable. With the linear discriminant model we later on predict the crime rate of the test data set.
# number of rows in the Boston dataset
n_boston <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind_boston <- sample(n_boston, size = n_boston * 0.8)
# create train set (crime variable stays in the data set)
boston_train <- boston_scaled[ind_boston,]
# create test set
boston_test <- boston_scaled[-ind_boston,]
knitr::kable(boston_test, caption = "Test data set with the crime variable") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center",) %>%
scroll_box(width = "100%", height = "300px") # the data frame head| zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | crime | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.0152978 | -0.8090878 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4157514 | -1.3601708 | 1.1815886 | low |
| 7 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3880270 | -0.0701592 | 0.8384142 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4263763 | -0.0312367 | 0.0399249 | med_low |
| 19 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.1793541 | -1.1359217 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | -0.7413783 | -0.1348628 | -0.2536457 | med_high |
| 20 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.7936533 | 0.0328645 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3754425 | -0.1922772 | -0.4711055 | med_high |
| 24 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 1.1163897 | 0.1425445 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4147656 | 1.0120256 | -0.8734060 | med_high |
| 31 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8135788 | 0.9067897 | 0.2079856 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0382932 | 1.3929213 | -1.0691198 | med_high |
| 36 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.5004637 | -0.0133185 | -0.2064589 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.4163335 | -0.3949946 | low |
| 37 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6314027 | -0.2548913 | -0.1981007 | -0.5224844 | -0.7668172 | 0.3438730 | 0.2287748 | -0.1740726 | -0.2753917 | med_low |
| 40 | 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 0.4417279 | -1.6616978 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4267049 | -1.1669222 | 0.8988910 | low |
| 43 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1645767 | -2.2016841 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.2924148 | -0.9582698 | 0.3008766 | med_low |
| 47 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.7096815 | -1.2531555 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.2096238 | -0.2753917 | med_low |
| 52 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.2414322 | -0.1980507 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4085221 | -0.4513423 | -0.2210268 | low |
| 54 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4079525 | -1.6759080 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -0.5913775 | 0.0942899 | low |
| 56 | 3.3717021 | -1.4455202 | -0.2723291 | -1.3090965 | 1.3725336 | -1.6581453 | 2.3277455 | -0.5224844 | -1.0812880 | -0.2566040 | 0.4299910 | -1.0983050 | 1.3990484 | low |
| 62 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.4534965 | 0.8819220 | 1.4358374 | -0.1779443 | -0.7371501 | 0.5748257 | 0.2344707 | 0.2502341 | -0.7103112 | med_low |
| 63 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.2438961 | -0.0275287 | 1.6291213 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.8294374 | -0.0361860 | med_low |
| 67 | 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | -0.7082582 | -1.3313114 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -0.3379138 | -0.3406296 | low |
| 70 | 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5687797 | -1.2638131 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | -0.5409648 | -0.1775348 | med_low |
| 76 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | 0.0019436 | -0.8375082 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.2908813 | -0.5199596 | -0.1231699 | med_low |
| 85 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.1485384 | -0.7309319 | 0.4674705 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4406159 | -0.4247356 | 0.1486548 | low |
| 94 | 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.1048002 | -1.4094674 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4343724 | -0.9022557 | 0.2682577 | low |
| 105 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1674232 | 0.7611355 | -0.6525317 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3945016 | -0.0452402 | -0.2645187 | med_low |
| 107 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6385190 | 0.8286338 | -0.7522606 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4271431 | 0.8411826 | -0.3297567 | med_low |
| 111 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1275722 | -0.5035693 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4032644 | 0.0485834 | -0.0905509 | med_low |
| 116 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5075800 | 0.6971898 | -0.6325385 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1288575 | 0.4350805 | -0.4602325 | med_low |
| 118 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.3752177 | 0.4982475 | -0.4975246 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4144370 | -0.3295117 | -0.3623756 | med_low |
| 128 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.8420438 | 0.9742880 | -0.9530004 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3881485 | 0.6353309 | -0.6885653 | med_high |
| 133 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.1243431 | 1.0417863 | -0.6969823 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3185937 | -0.2146828 | 0.0507979 | med_high |
| 149 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.5636316 | 0.8961321 | -1.0758569 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0034610 | 2.1939227 | -0.5145975 | med_high |
| 155 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.2215067 | 0.9742880 | -0.9714740 | -0.5224844 | -0.0310742 | -1.7347012 | -0.3905371 | 0.3454580 | -0.6015814 | med_high |
| 156 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.1887719 | 0.4982475 | -0.9733261 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9428165 | 0.3314545 | -0.7538032 | med_high |
| 161 | -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | -0.0492934 | 0.8535016 | -0.9482040 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1944691 | -1.0016807 | 0.4857174 | med_high |
| 165 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.6129005 | 0.8250813 | -0.6520568 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4210091 | -0.1418645 | 0.0181789 | med_high |
| 166 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.2613577 | 0.8677118 | -0.7178779 | -0.5224844 | -0.0310742 | -1.7347012 | -1.2762386 | -0.3981289 | 0.2682577 | med_high |
| 171 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5830122 | 0.9245525 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -0.7052317 | 0.2488337 | -0.5580894 | med_high |
| 174 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.1869661 | 0.5515356 | -0.5455370 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4252810 | -0.5059560 | 0.1160358 | med_low |
| 179 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.8188892 | 0.2069391 | -0.4177891 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3789476 | -0.8028307 | 0.8010341 | low |
| 180 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.9896793 | -0.3614676 | -0.4587729 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -1.0660969 | 1.5947622 | low |
| 182 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.2001579 | -0.2264710 | -0.5685221 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.4485416 | 1.4860323 | low |
| 188 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.7078757 | -0.9760573 | -0.0030596 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4074267 | -0.8364391 | 1.0293668 | low |
| 192 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.6466760 | -1.3419691 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3618601 | -1.1151092 | 0.8662720 | low |
| 198 | 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.1704320 | -1.1359217 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | -0.0258945 | -0.5661712 | 0.8445260 | low |
| 206 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.5602402 | -1.6439351 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | -0.2496916 | 0.0073060 | med_low |
| 207 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.0588736 | -0.5710675 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4183803 | -0.2356881 | 0.2030197 | med_low |
| 212 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.2533631 | 0.7114000 | -0.0617572 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4224331 | 1.5861699 | -0.3515026 | med_high |
| 213 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.6797932 | -0.5248845 | -0.0676459 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3753329 | 0.4728900 | -0.0144400 | med_low |
| 216 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.1460744 | -0.9298742 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4047979 | -0.4457409 | 0.2682577 | med_low |
| 223 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.8459310 | 0.3241729 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3693085 | -0.3813247 | 0.5400824 | med_high |
| 232 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.6045233 | 0.2957526 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2132208 | -1.0366895 | 0.9967478 | med_high |
| 233 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.9210298 | 0.1678611 | 0.0205904 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3202367 | -1.4259873 | 2.0840466 | med_high |
| 240 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.4573837 | -0.9369793 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2969057 | -0.7398148 | 0.0834169 | med_low |
| 245 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9843688 | 0.2815424 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1732405 | -0.0214342 | -0.5363434 | med_low |
| 249 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2111614 | -0.6918540 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1975573 | -0.4387391 | 0.2138927 | med_low |
| 256 | 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.5815890 | -1.7576164 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.4217758 | -0.4765487 | -0.1775348 | low |
| 257 | 3.3717021 | -1.0767345 | -0.2723291 | -1.3867646 | 1.6642999 | -1.2211827 | 1.2067460 | -0.7521778 | -0.9744866 | -1.1804147 | 0.3249467 | -1.3363648 | 2.3341253 | low |
| 262 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.7582344 | 0.7398203 | -0.7860734 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3471824 | -0.7552187 | 2.2362684 | med_high |
| 268 | 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 2.8640998 | -0.0559490 | -0.6522468 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3052304 | -0.7300124 | 2.9865046 | med_high |
| 270 | 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | -0.5189660 | -0.2513388 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3797143 | 0.1396062 | -0.1992808 | med_low |
| 271 | 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.6100540 | -0.9405319 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3502494 | 0.0485834 | -0.1557889 | med_high |
| 285 | 3.3717021 | -1.1904313 | -0.2723291 | -1.3349859 | 1.1433903 | -1.6972232 | 1.6679681 | -0.9818712 | -0.7312167 | -1.4575580 | 0.4167372 | -0.6725979 | 1.0511128 | low |
| 287 | 2.9429307 | -1.3668070 | -0.2723291 | -1.4644327 | -0.0777584 | -1.3171013 | 2.5141909 | -0.9818712 | -0.9922868 | -0.1180323 | -0.1651137 | 0.0387809 | -0.2645187 | low |
| 292 | 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 1.2287853 | -1.4520979 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.2733490 | 1.6056352 | low |
| 302 | 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.4346117 | -1.0009251 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.4280193 | -0.4415399 | -0.0579320 | low |
| 309 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4986579 | 0.4946949 | -0.2267846 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -1.1361145 | 0.0290519 | med_high |
| 315 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4018769 | 0.6652169 | -0.0915333 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4273621 | -0.4723476 | 0.1377818 | med_high |
| 319 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.1385756 | -0.0488439 | -0.1246813 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4221044 | -0.3211096 | 0.0616709 | med_high |
| 329 | -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.5929750 | -1.5195961 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2822280 | -0.3757233 | -0.3515026 | low |
| 337 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5915517 | -0.7913251 | 0.6819824 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.3995293 | -0.3297567 | low |
| 338 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5545472 | -0.3188371 | 0.8642962 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4177230 | -0.2931025 | -0.4384865 | low |
| 340 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4264547 | -0.8232980 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4079314 | -0.3841216 | low |
| 341 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4506500 | -0.3579151 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4709472 | -0.4167406 | low |
| 347 | -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.5502775 | -0.5781726 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.0869268 | 0.0023717 | -0.5798354 | low |
| 357 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.1033769 | 1.0240236 | -0.7944316 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2306369 | 0.6927453 | -0.5145975 | high |
| 359 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.2243532 | 0.5266678 | -0.5092547 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4245142 | -0.1642701 | 0.0181789 | high |
| 360 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.2457019 | 0.4520644 | -0.6106931 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3731422 | 0.0023717 | 0.0073060 | high |
| 373 | -0.4872402 | 1.0149946 | 3.6647712 | 0.9777978 | -0.5830122 | 0.7469254 | -1.2658165 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0963256 | -0.5283617 | 2.9865046 | high |
| 374 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -1.9621417 | 1.1163897 | -1.2446360 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.0971497 | -0.9495170 | high |
| 378 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.7249547 | 1.0737592 | -1.1573496 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.2024734 | -1.0038819 | high |
| 393 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.7771192 | 1.0098134 | -0.9616911 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.8242297 | -1.3953095 | high |
| 403 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1698871 | 1.1163897 | -1.0239029 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2128922 | 1.0722407 | -1.1343578 | high |
| 413 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -2.3578052 | 1.1163897 | -1.0643168 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5914839 | 3.0411357 | -0.5037245 | high |
| 414 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -1.6077524 | 1.1163897 | -1.0474579 | 1.6596029 | 1.5294129 | 0.8057784 | -1.5959718 | 1.0400326 | -0.6776923 | high |
| 423 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.9060900 | 0.6758745 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7133373 | 0.2026221 | -0.1884078 | high |
| 426 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.5531240 | 0.9529728 | -0.8953952 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8227126 | 1.6435843 | -1.5475313 | high |
| 434 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.2154311 | 0.6865322 | -0.7024911 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8094026 | 0.4994967 | -0.8951520 | high |
| 435 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1090699 | 0.9387626 | -0.7469417 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8045831 | 0.3524598 | -1.1778497 | high |
| 436 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.4901184 | 0.9245525 | -0.7932444 | 1.6596029 | 1.5294129 | 0.8057784 | -2.7035916 | 1.4867449 | -0.9930089 | high |
| 446 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2481659 | 0.9316575 | -0.8582106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.4351771 | 1.5861699 | -1.1669767 | high |
| 452 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5271229 | 1.0524439 | -0.6837801 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0151600 | 0.7109499 | -0.7972951 | high |
| 453 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0175994 | 0.8250813 | -0.6776064 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3112548 | 0.6465337 | -0.6994382 | high |
| 458 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4961940 | 0.4165390 | -0.4824229 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8684983 | 0.6003221 | -0.9821359 | high |
| 468 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.4008362 | 0.9209999 | -0.5958763 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2780445 | 1.2136763 | -0.3732486 | high |
| 469 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.5104265 | 0.0861526 | -0.4210659 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1321648 | 0.7669640 | -0.3732486 | high |
| 471 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.1674232 | 0.5479830 | -0.3617035 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5092992 | -0.2862647 | high |
| 478 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -1.3956881 | 1.0204711 | -0.8046419 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0788000 | 1.7164026 | -1.1452307 | high |
| 485 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5887052 | -0.9476370 | -0.0337381 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1539623 | 0.0961953 | -0.2101538 | med_high |
| 487 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.2428554 | 0.3987763 | -0.1183177 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3943920 | 0.3258531 | -0.3732486 | high |
| 489 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.1822006 | 0.8570542 | -0.9375187 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4207900 | 0.7571615 | -0.7972951 | med_low |
| 490 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.2391306 | 1.0559965 | -0.9686246 | -0.6373311 | 1.7964164 | 0.7595879 | -0.1382776 | 1.5847695 | -1.6888801 | med_low |
| 491 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.6959939 | 1.0453389 | -0.9367114 | -0.6373311 | 1.7964164 | 0.7595879 | -0.4189067 | 2.3843705 | -1.5692773 | med_low |
| 495 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.5104265 | -0.9227692 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.1312041 | 0.2138927 | med_high |
| 502 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.4388814 | 0.0186544 | -0.6251775 | -0.9818712 | -0.8024176 | 1.1753027 | 0.3868341 | -0.4177339 | -0.0144400 | low |
# save the correct classes from test data
correct_classes <- boston_test$crime
correct_classes## [1] low med_low med_high med_high med_high med_high low
## [8] med_low low med_low med_low low low low
## [15] med_low med_low low med_low med_low low low
## [22] med_low med_low med_low med_low med_low med_high med_high
## [29] med_high med_high med_high med_high med_high med_high med_high
## [36] med_low low low low low low low
## [43] med_low med_low med_high med_low med_low med_high med_high
## [50] med_high med_low med_low med_low low low med_high
## [57] med_high med_low med_high low low low low
## [64] med_high med_high med_high low low low low
## [71] low low high high high high high
## [78] high high high high high high high
## [85] high high high high high high high
## [92] high high high high med_high high med_low
## [99] med_low med_low med_high low
## Levels: low med_low med_high high# remove the crime variable from test data set / so the prediction can be done later!
boston_test <- dplyr::select(boston_test, -crime)
knitr::kable(boston_test, caption = "Test data without the crime variable") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center",) %>%
scroll_box(width = "100%", height = "300px") # the data frame head| zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.0152978 | -0.8090878 | 1.0766711 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4157514 | -1.3601708 | 1.1815886 |
| 7 | 0.0487240 | -0.4761823 | -0.2723291 | -0.2648919 | -0.3880270 | -0.0701592 | 0.8384142 | -0.5224844 | -0.5769480 | -1.5037485 | 0.4263763 | -0.0312367 | 0.0399249 |
| 19 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -1.1793541 | -1.1359217 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | -0.7413783 | -0.1348628 | -0.2536457 |
| 20 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.7936533 | 0.0328645 | 0.0006921 | -0.6373311 | -0.6006817 | 1.1753027 | 0.3754425 | -0.1922772 | -0.4711055 |
| 24 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.6712537 | 1.1163897 | 0.1425445 | -0.6373311 | -0.6006817 | 1.1753027 | 0.4147656 | 1.0120256 | -0.8734060 |
| 31 | -0.4872402 | -0.4368257 | -0.2723291 | -0.1440749 | -0.8135788 | 0.9067897 | 0.2079856 | -0.6373311 | -0.6006817 | 1.1753027 | 0.0382932 | 1.3929213 | -1.0691198 |
| 36 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.5004637 | -0.0133185 | -0.2064589 | -0.5224844 | -0.7668172 | 0.3438730 | 0.4406159 | -0.4163335 | -0.3949946 |
| 37 | -0.4872402 | -0.7545936 | -0.2723291 | -0.4806367 | -0.6314027 | -0.2548913 | -0.1981007 | -0.5224844 | -0.7668172 | 0.3438730 | 0.2287748 | -0.1740726 | -0.2753917 |
| 40 | 2.7285450 | -1.1933466 | -0.2723291 | -1.0933517 | 0.4417279 | -1.6616978 | 0.7627153 | -0.7521778 | -0.9270193 | -0.0718418 | 0.4267049 | -1.1669222 | 0.8988910 |
| 43 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.1645767 | -2.2016841 | 0.9145880 | -0.7521778 | -1.0397541 | -0.2566040 | 0.2924148 | -0.9582698 | 0.3008766 |
| 47 | -0.4872402 | -0.6161168 | -0.2723291 | -0.9207559 | -0.7096815 | -1.2531555 | 0.6199131 | -0.7521778 | -1.0397541 | -0.2566040 | 0.4406159 | 0.2096238 | -0.2753917 |
| 52 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.2414322 | -0.1980507 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4085221 | -0.4513423 | -0.2210268 |
| 54 | 0.4131797 | -0.8012385 | -0.2723291 | -0.9984241 | -0.4079525 | -1.6759080 | 1.4340328 | -0.6373311 | -0.9804200 | -0.7646999 | 0.4406159 | -0.5913775 | 0.0942899 |
| 56 | 3.3717021 | -1.4455202 | -0.2723291 | -1.3090965 | 1.3725336 | -1.6581453 | 2.3277455 | -0.5224844 | -1.0812880 | -0.2566040 | 0.4299910 | -1.0983050 | 1.3990484 |
| 62 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | -0.4534965 | 0.8819220 | 1.4358374 | -0.1779443 | -0.7371501 | 0.5748257 | 0.2344707 | 0.2502341 | -0.7103112 |
| 63 | 0.5846882 | -0.8755787 | -0.2723291 | -0.8776070 | 0.2438961 | -0.0275287 | 1.6291213 | -0.1779443 | -0.7371501 | 0.5748257 | 0.4406159 | -0.8294374 | -0.0361860 |
| 67 | 2.9429307 | -1.1321252 | -0.2723291 | -1.3522454 | -0.7082582 | -1.3313114 | 1.3375333 | -0.6373311 | -0.4226793 | -1.0880337 | 0.4406159 | -0.3379138 | -0.3406296 |
| 70 | 0.0487240 | -0.7385595 | -0.2723291 | -1.2573178 | -0.5687797 | -1.2638131 | 1.2836322 | -0.6373311 | -0.3752120 | 0.2053014 | 0.4406159 | -0.5409648 | -0.1775348 |
| 76 | -0.4872402 | 0.2468126 | -0.2723291 | -1.0156836 | 0.0019436 | -0.8375082 | 0.3360184 | -0.5224844 | -0.0607412 | 0.1129203 | 0.2908813 | -0.5199596 | -0.1231699 |
| 85 | -0.4872402 | -0.9688683 | -0.2723291 | -0.9121262 | 0.1485384 | -0.7309319 | 0.4674705 | -0.7521778 | -0.9566863 | 0.0205393 | 0.4406159 | -0.4247356 | 0.1486548 |
| 94 | 0.7133196 | 0.5689534 | -0.2723291 | -0.7826793 | -0.1048002 | -1.4094674 | -0.0613298 | -0.6373311 | -0.8202179 | -0.1180323 | 0.4343724 | -0.9022557 | 0.2682577 |
| 105 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1674232 | 0.7611355 | -0.6525317 | -0.5224844 | -0.1438090 | 1.1291122 | 0.3945016 | -0.0452402 | -0.2645187 |
| 107 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.6385190 | 0.8286338 | -0.7522606 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4271431 | 0.8411826 | -0.3297567 |
| 111 | -0.4872402 | -0.3756044 | -0.2723291 | -0.2994111 | -0.1275722 | -0.5035693 | -0.4830877 | -0.5224844 | -0.1438090 | 1.1291122 | 0.4032644 | 0.0485834 | -0.0905509 |
| 116 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.5075800 | 0.6971898 | -0.6325385 | -0.4076377 | 0.1409947 | -0.3027945 | -0.1288575 | 0.4350805 | -0.4602325 |
| 118 | -0.4872402 | -0.1642450 | -0.2723291 | -0.0664067 | -0.3752177 | 0.4982475 | -0.4975246 | -0.4076377 | 0.1409947 | -0.3027945 | 0.4144370 | -0.3295117 | -0.3623756 |
| 128 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | -0.8420438 | 0.9742880 | -0.9530004 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3881485 | 0.6353309 | -0.6885653 |
| 133 | -0.4872402 | 1.5674443 | -0.2723291 | 0.5980871 | 0.1243431 | 1.0417863 | -0.6969823 | -0.6373311 | 0.1706618 | 1.2676838 | 0.3185937 | -0.2146828 | 0.0507979 |
| 149 | -0.4872402 | 1.2307270 | -0.2723291 | 2.7296452 | -1.5636316 | 0.8961321 | -1.0758569 | -0.5224844 | -0.0310742 | -1.7347012 | 0.0034610 | 2.1939227 | -0.5145975 |
| 155 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.2215067 | 0.9742880 | -0.9714740 | -0.5224844 | -0.0310742 | -1.7347012 | -0.3905371 | 0.3454580 | -0.6015814 |
| 156 | -0.4872402 | 1.2307270 | 3.6647712 | 2.7296452 | -0.1887719 | 0.4982475 | -0.9733261 | -0.5224844 | -0.0310742 | -1.7347012 | -2.9428165 | 0.3314545 | -0.7538032 |
| 161 | -0.4872402 | 1.2307270 | 3.6647712 | 0.4341211 | -0.0492934 | 0.8535016 | -0.9482040 | -0.5224844 | -0.0310742 | -1.7347012 | -0.1944691 | -1.0016807 | 0.4857174 |
| 165 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.6129005 | 0.8250813 | -0.6520568 | -0.5224844 | -0.0310742 | -1.7347012 | 0.4210091 | -0.1418645 | 0.0181789 |
| 166 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.2613577 | 0.8677118 | -0.7178779 | -0.5224844 | -0.0310742 | -1.7347012 | -1.2762386 | -0.3981289 | 0.2682577 |
| 171 | -0.4872402 | 1.2307270 | -0.2723291 | 0.4341211 | -0.5830122 | 0.9245525 | -0.6502047 | -0.5224844 | -0.0310742 | -1.7347012 | -0.7052317 | 0.2488337 | -0.5580894 |
| 174 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.1869661 | 0.5515356 | -0.5455370 | -0.5224844 | -0.6659492 | -0.8570810 | 0.4252810 | -0.5059560 | 0.1160358 |
| 179 | -0.4872402 | -1.0330050 | -0.2723291 | -0.3857090 | 0.8188892 | 0.2069391 | -0.4177891 | -0.5224844 | -0.6659492 | -0.8570810 | 0.3789476 | -0.8028307 | 0.8010341 |
| 180 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | 0.9896793 | -0.3614676 | -0.4587729 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -1.0660969 | 1.5947622 |
| 182 | -0.4872402 | -1.2647715 | -0.2723291 | -0.5755644 | -0.2001579 | -0.2264710 | -0.5685221 | -0.7521778 | -1.2770905 | -0.3027945 | 0.4406159 | -0.4485416 | 1.4860323 |
| 188 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.7078757 | -0.9760573 | -0.0030596 | -0.5224844 | -0.0607412 | -1.5037485 | 0.4074267 | -0.8364391 | 1.0293668 |
| 192 | 1.4422310 | -1.1219217 | -0.2723291 | -1.0156836 | 0.6466760 | -1.3419691 | 1.2749890 | -0.5224844 | -0.0607412 | -1.5037485 | 0.3618601 | -1.1151092 | 0.8662720 |
| 198 | 2.9429307 | -1.4017907 | -0.2723291 | -1.3004667 | 1.1704320 | -1.1359217 | 1.6687754 | -0.8670245 | -0.4701466 | -2.7047025 | -0.0258945 | -0.5661712 | 0.8445260 |
| 206 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.5602402 | -1.6439351 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4406159 | -0.2496916 | 0.0073060 |
| 207 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | 0.0588736 | -0.5710675 | 0.2658758 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4183803 | -0.2356881 | 0.2030197 |
| 212 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -1.2533631 | 0.7114000 | -0.0617572 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4224331 | 1.5861699 | -0.3515026 |
| 213 | -0.4872402 | -0.0797012 | 3.6647712 | -0.5669346 | -0.6797932 | -0.5248845 | -0.0676459 | -0.6373311 | -0.7786840 | 0.0667298 | 0.3753329 | 0.4728900 | -0.0144400 |
| 216 | -0.4872402 | -0.0797012 | -0.2723291 | -0.5669346 | -0.1460744 | -0.9298742 | 0.0714046 | -0.6373311 | -0.7786840 | 0.0667298 | 0.4047979 | -0.4457409 | 0.2682577 |
| 223 | -0.4872402 | -0.7196100 | 3.6647712 | -0.4115983 | 0.8459310 | 0.3241729 | -0.2483451 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3693085 | -0.3813247 | 0.5400824 |
| 232 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4374877 | 1.6045233 | 0.2957526 | -0.0586703 | -0.1779443 | -0.6006817 | -0.4875567 | 0.2132208 | -1.0366895 | 0.9967478 |
| 233 | -0.4872402 | -0.7196100 | -0.2723291 | -0.4115983 | 2.9210298 | 0.1678611 | 0.0205904 | -0.1779443 | -0.6006817 | -0.4875567 | 0.3202367 | -1.4259873 | 2.0840466 |
| 240 | 0.7990739 | -0.9047317 | -0.2723291 | -1.0933517 | 0.4573837 | -0.9369793 | 1.1373158 | -0.4076377 | -0.6422155 | -0.8570810 | 0.2969057 | -0.7398148 | 0.0834169 |
| 245 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | -0.9843688 | 0.2815424 | 1.9755128 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1732405 | -0.0214342 | -0.5363434 |
| 249 | 0.4560568 | -0.7691701 | -0.2723291 | -1.0674624 | 0.2111614 | -0.6918540 | 1.9145357 | -0.2927910 | -0.4642132 | 0.2976825 | 0.1975573 | -0.4387391 | 0.2138927 |
| 256 | 2.9429307 | -1.0927687 | -0.2723291 | -1.4040242 | -0.5815890 | -1.7576164 | 2.5764502 | -0.9818712 | -0.5532144 | -0.9494620 | 0.4217758 | -0.4765487 | -0.1775348 |
| 257 | 3.3717021 | -1.0767345 | -0.2723291 | -1.3867646 | 1.6642999 | -1.2211827 | 1.2067460 | -0.7521778 | -0.9744866 | -1.1804147 | 0.3249467 | -1.3363648 | 2.3341253 |
| 262 | 0.3703025 | -1.0446662 | -0.2723291 | 0.7965722 | 1.7582344 | 0.7398203 | -0.7860734 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3471824 | -0.7552187 | 2.2362684 |
| 268 | 0.3703025 | -1.0446662 | -0.2723291 | 0.1752274 | 2.8640998 | -0.0559490 | -0.6522468 | -0.5224844 | -0.8558183 | -2.5199404 | 0.3052304 | -0.7300124 | 2.9865046 |
| 270 | 0.3703025 | -0.6088285 | 3.6647712 | -0.7826793 | -0.5189660 | -0.2513388 | 0.0581549 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3797143 | 0.1396062 | -0.1992808 |
| 271 | 0.3703025 | -0.6088285 | -0.2723291 | -0.7826793 | -0.6100540 | -0.9405319 | 0.3010658 | -0.7521778 | -1.0990882 | 0.0667298 | 0.3502494 | 0.0485834 | -0.1557889 |
| 285 | 3.3717021 | -1.1904313 | -0.2723291 | -1.3349859 | 1.1433903 | -1.6972232 | 1.6679681 | -0.9818712 | -0.7312167 | -1.4575580 | 0.4167372 | -0.6725979 | 1.0511128 |
| 287 | 2.9429307 | -1.3668070 | -0.2723291 | -1.4644327 | -0.0777584 | -1.3171013 | 2.5141909 | -0.9818712 | -0.9922868 | -0.1180323 | -0.1651137 | 0.0387809 | -0.2645187 |
| 292 | 2.9429307 | -0.9018164 | -0.2723291 | -1.2400582 | 1.2287853 | -1.4520979 | 0.6276540 | -0.6373311 | -0.9685531 | 0.3438730 | 0.4406159 | -1.2733490 | 1.6056352 |
| 302 | 0.9705825 | -0.7356442 | -0.2723291 | -1.0502028 | 0.4346117 | -1.0009251 | 0.8057412 | -0.2927910 | -0.4701466 | -1.0880337 | 0.4280193 | -0.4415399 | -0.0579320 |
| 309 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4986579 | 0.4946949 | -0.2267846 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4406159 | -1.1361145 | 0.0290519 |
| 315 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.4018769 | 0.6652169 | -0.0915333 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4273621 | -0.4723476 | 0.1377818 |
| 319 | -0.4872402 | -0.1802792 | -0.2723291 | -0.0922961 | 0.1385756 | -0.0488439 | -0.1246813 | -0.6373311 | -0.6184819 | -0.0256513 | 0.4221044 | -0.3211096 | 0.0616709 |
| 329 | -0.4872402 | -1.1510747 | -0.2723291 | -0.8171985 | -0.5929750 | -1.5195961 | 0.6741466 | -0.6373311 | 0.1291279 | -0.7185093 | 0.2822280 | -0.3757233 | -0.3515026 |
| 337 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5915517 | -0.7913251 | 0.6819824 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.3995293 | -0.3297567 |
| 338 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.5545472 | -0.3188371 | 0.8642962 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4177230 | -0.2931025 | -0.4384865 |
| 340 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4264547 | -0.8232980 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4079314 | -0.3841216 |
| 341 | -0.4872402 | -0.8668328 | -0.2723291 | -0.3425600 | -0.4506500 | -0.3579151 | 0.4830472 | -0.5224844 | -1.0931548 | 0.8057784 | 0.4406159 | -0.4709472 | -0.4167406 |
| 347 | -0.4872402 | -0.9834448 | -0.2723291 | -0.9725347 | -0.5502775 | -0.5781726 | 2.0033894 | -0.7521778 | -0.3336782 | 0.1591109 | 0.0869268 | 0.0023717 | -0.5798354 |
| 357 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.1033769 | 1.0240236 | -0.7944316 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2306369 | 0.6927453 | -0.5145975 |
| 359 | -0.4872402 | 1.0149946 | 3.6647712 | 1.8580364 | -0.2243532 | 0.5266678 | -0.5092547 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4245142 | -0.1642701 | 0.0181789 |
| 360 | -0.4872402 | 1.0149946 | -0.2723291 | 1.8580364 | -0.2457019 | 0.4520644 | -0.6106931 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3731422 | 0.0023717 | 0.0073060 |
| 373 | -0.4872402 | 1.0149946 | 3.6647712 | 0.9777978 | -0.5830122 | 0.7469254 | -1.2658165 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0963256 | -0.5283617 | 2.9865046 |
| 374 | -0.4872402 | 1.0149946 | -0.2723291 | 0.9777978 | -1.9621417 | 1.1163897 | -1.2446360 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 3.0971497 | -0.9495170 |
| 378 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0036872 | 0.7249547 | 1.0737592 | -1.1573496 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.2024734 | -1.0038819 |
| 393 | -0.4872402 | 1.0149946 | -0.2723291 | 1.2539511 | -1.7771192 | 1.0098134 | -0.9616911 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 1.8242297 | -1.3953095 |
| 403 | -0.4872402 | 1.0149946 | -0.2723291 | 1.1935426 | 0.1698871 | 1.1163897 | -1.0239029 | 1.6596029 | 1.5294129 | 0.8057784 | 0.2128922 | 1.0722407 | -1.1343578 |
| 413 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -2.3578052 | 1.1163897 | -1.0643168 | 1.6596029 | 1.5294129 | 0.8057784 | -3.5914839 | 3.0411357 | -0.5037245 |
| 414 | -0.4872402 | 1.0149946 | -0.2723291 | 0.3650828 | -1.6077524 | 1.1163897 | -1.0474579 | 1.6596029 | 1.5294129 | 0.8057784 | -1.5959718 | 1.0400326 | -0.6776923 |
| 423 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -0.9060900 | 0.6758745 | -0.8756394 | 1.6596029 | 1.5294129 | 0.8057784 | -0.7133373 | 0.2026221 | -0.1884078 |
| 426 | -0.4872402 | 1.0149946 | -0.2723291 | 1.0727255 | -0.5531240 | 0.9529728 | -0.8953952 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8227126 | 1.6435843 | -1.5475313 |
| 434 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.2154311 | 0.6865322 | -0.7024911 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8094026 | 0.4994967 | -0.8951520 |
| 435 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.1090699 | 0.9387626 | -0.7469417 | 1.6596029 | 1.5294129 | 0.8057784 | -2.8045831 | 0.3524598 | -1.1778497 |
| 436 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.4901184 | 0.9245525 | -0.7932444 | 1.6596029 | 1.5294129 | 0.8057784 | -2.7035916 | 1.4867449 | -0.9930089 |
| 446 | -0.4872402 | 1.0149946 | -0.2723291 | 1.5991427 | 0.2481659 | 0.9316575 | -0.8582106 | 1.6596029 | 1.5294129 | 0.8057784 | -3.4351771 | 1.5861699 | -1.1669767 |
| 452 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.5271229 | 1.0524439 | -0.6837801 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0151600 | 0.7109499 | -0.7972951 |
| 453 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | 0.0175994 | 0.8250813 | -0.6776064 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3112548 | 0.6465337 | -0.6994382 |
| 458 | -0.4872402 | 1.0149946 | -0.2723291 | 1.3661384 | -0.4961940 | 0.4165390 | -0.4824229 | 1.6596029 | 1.5294129 | 0.8057784 | -3.8684983 | 0.6003221 | -0.9821359 |
| 468 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2528955 | -0.4008362 | 0.9209999 | -0.5958763 | 1.6596029 | 1.5294129 | 0.8057784 | -0.2780445 | 1.2136763 | -0.3732486 |
| 469 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.5104265 | 0.0861526 | -0.4210659 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1321648 | 0.7669640 | -0.3732486 |
| 471 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2183763 | -0.1674232 | 0.5479830 | -0.3617035 | 1.6596029 | 1.5294129 | 0.8057784 | 0.4406159 | 0.5092992 | -0.2862647 |
| 478 | -0.4872402 | 1.0149946 | -0.2723291 | 0.5117892 | -1.3956881 | 1.0204711 | -0.8046419 | 1.6596029 | 1.5294129 | 0.8057784 | -0.0788000 | 1.7164026 | -1.1452307 |
| 485 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.5887052 | -0.9476370 | -0.0337381 | 1.6596029 | 1.5294129 | 0.8057784 | 0.1539623 | 0.0961953 | -0.2101538 |
| 487 | -0.4872402 | 1.0149946 | -0.2723291 | 0.2442657 | -0.2428554 | 0.3987763 | -0.1183177 | 1.6596029 | 1.5294129 | 0.8057784 | 0.3943920 | 0.3258531 | -0.3732486 |
| 489 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.1822006 | 0.8570542 | -0.9375187 | -0.6373311 | 1.7964164 | 0.7595879 | 0.4207900 | 0.7571615 | -0.7972951 |
| 490 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.2391306 | 1.0559965 | -0.9686246 | -0.6373311 | 1.7964164 | 0.7595879 | -0.1382776 | 1.5847695 | -1.6888801 |
| 491 | -0.4872402 | 2.4201701 | -0.2723291 | 0.4686402 | -1.6959939 | 1.0453389 | -0.9367114 | -0.6373311 | 1.7964164 | 0.7595879 | -0.4189067 | 2.3843705 | -1.5692773 |
| 495 | -0.4872402 | -0.2108898 | -0.2723291 | 0.2615253 | -0.5104265 | -0.9227692 | -0.6711953 | -0.4076377 | -0.1022751 | 0.3438730 | 0.4406159 | 0.1312041 | 0.2138927 |
| 502 | -0.4872402 | 0.1156240 | -0.2723291 | 0.1579678 | 0.4388814 | 0.0186544 | -0.6251775 | -0.9818712 | -0.8024176 | 1.1753027 | 0.3868341 | -0.4177339 | -0.0144400 |
1.3.4. Linear discriminant analysis of the training data set
Here we run now the LDA ananlysis on the ‘boston_train’ data set. So we make a classification of the “crime” variable of our training data set.
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = boston_train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = boston_train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2450495 0.2500000 0.2500000 0.2549505
##
## Group means:
## zn indus chas nox rm
## low 0.96776729 -0.8980913 -0.07348562 -0.8791063 0.50672302
## med_low -0.07184139 -0.3303595 -0.03844192 -0.5777090 -0.06234221
## med_high -0.38365582 0.1784617 0.11748284 0.3922538 0.12142617
## high -0.48724019 1.0170891 -0.11943197 1.0609119 -0.39758898
## age dis rad tax ptratio black
## low -0.8717059 0.8832840 -0.6941744 -0.7255230 -0.4637615 0.3823775
## med_low -0.3785268 0.3804356 -0.5531860 -0.5303619 -0.1175750 0.3305797
## med_high 0.4228703 -0.3713792 -0.4042264 -0.3041286 -0.2524880 0.1095973
## high 0.8109401 -0.8704238 1.6384176 1.5142626 0.7811136 -0.7552212
## lstat medv
## low -0.80551822 0.55721555
## med_low -0.17588891 0.06296274
## med_high 0.01905118 0.17739824
## high 0.86812919 -0.71685614
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.060099250 0.73620470 -0.9855179
## indus 0.008064383 -0.12780753 0.3095989
## chas -0.101073981 -0.04681437 0.0461922
## nox 0.374799602 -0.85050153 -1.3408260
## rm -0.089856978 -0.05820899 -0.2142469
## age 0.258592866 -0.23699091 -0.1745051
## dis -0.057768768 -0.26744922 0.1558164
## rad 3.087398715 1.05894467 0.1457907
## tax 0.118815758 -0.14944306 0.3254184
## ptratio 0.121013500 -0.04833078 -0.3535688
## black -0.128757895 0.03061410 0.1343696
## lstat 0.206677615 -0.24731102 0.3946104
## medv 0.182532860 -0.43817126 -0.1512821
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9521 0.0361 0.0117# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(boston_train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes, main = "LDA biplot of Boston training data")
lda.arrows(lda.fit, myscale = 2)
The linear discriminant analysis shows the classification of the “crime” variable. In the LDA plot we can see the separation of the different crime rates.
The LDA creates a value which characterize which variable discriminate most of the variables. So now we have performed the LDA on the training data set. Now we use the LDA outcome to perform a prediction on the crime rate on the testing data set.
1.3.5. Prediction of crime classes on the test data set
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = boston_test)
lda.pred$class## [1] med_low med_low med_low med_low med_high med_high med_low
## [8] med_low low med_low med_low med_low low low
## [15] med_low med_low low med_low med_low med_low low
## [22] med_high med_high med_low med_high med_low med_high med_high
## [29] med_high med_high med_high med_high med_high med_high med_high
## [36] med_low med_low med_low med_low low low low
## [43] med_low med_low med_low med_low med_low med_low med_high
## [50] med_high low med_low med_low low low med_high
## [57] med_high med_low low low low low low
## [64] med_low med_high med_low med_low med_low med_low med_low
## [71] med_low med_low high high high high high
## [78] high high high high high high high
## [85] high high high high high high high
## [92] high high high high high high med_high
## [99] med_high med_high med_high med_high
## Levels: low med_low med_high high# cross tabulate the results
pred_table <- table(correct = correct_classes, predicted = lda.pred$class)
knitr::kable(pred_table, align = "c", caption="Pediction of test data" ) %>% kable_styling(bootstrap_options = "striped", full_width = F, position = "center") %>% add_header_above(c(" " = 1, "Predicted crime rate" = 4)) %>% pack_rows("Correct rate", 1, 4)| low | med_low | med_high | high | |
|---|---|---|---|---|
| Correct rate | ||||
| low | 14 | 13 | 1 | 0 |
| med_low | 1 | 18 | 6 | 0 |
| med_high | 1 | 6 | 17 | 1 |
| high | 0 | 0 | 0 | 24 |
It seems that the prediction is maybe not the best using the linear descriminant analysis.
The prediction showed not so good results on the lower crime rate predictions. Results show that the crime rate was predicted higher than it actually is (see in predicted med_low of and med_high which are actually low rates or med_low rates). In the medium crime level it also wrongly predicted quite a number of values, but on the high crime rates the predictions is quite good with just on value wrongly predicted.
1.3.5. Reload and standardize the Boston dataset and calculate the distances
Now we are going to perform a data clustering. This is different from classifications because in classifications the classes are defined before (in our case the different crime rate values from low to high). Here we analyise the data and through this analysis the clustering will show us were the data is different.
So here we are going to reload the orgininal ‘Boston’ data set and scale the data again to a new data frame ‘boston_scaled2’. Then we calculate the euclidian distances between the obsvervations. Then the k-means are calculated and
#load boston_scaled data set
data(Boston) # load the Boston data set
# scale the Boston data set again - named boston_scaled2
boston_scaled2 <- scale(Boston)
knitr::kable(head(boston_scaled2), caption = "Scaled Boston data set" ) %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center",) %>%
scroll_box(width = "100%", height = "300px") # the data frame head| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | black | lstat | medv |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| -0.4193669 | 0.2845483 | -1.2866362 | -0.2723291 | -0.1440749 | 0.4132629 | -0.1198948 | 0.140075 | -0.9818712 | -0.6659492 | -1.4575580 | 0.4406159 | -1.0744990 | 0.1595278 |
| -0.4169267 | -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 0.1940824 | 0.3668034 | 0.556609 | -0.8670245 | -0.9863534 | -0.3027945 | 0.4406159 | -0.4919525 | -0.1014239 |
| -0.4169290 | -0.4872402 | -0.5927944 | -0.2723291 | -0.7395304 | 1.2814456 | -0.2655490 | 0.556609 | -0.8670245 | -0.9863534 | -0.3027945 | 0.3960351 | -1.2075324 | 1.3229375 |
| -0.4163384 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.0152978 | -0.8090878 | 1.076671 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4157514 | -1.3601708 | 1.1815886 |
| -0.4120741 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 1.2273620 | -0.5106743 | 1.076671 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4406159 | -1.0254866 | 1.4860323 |
| -0.4166314 | -0.4872402 | -1.3055857 | -0.2723291 | -0.8344581 | 0.2068916 | -0.3508100 | 1.076671 | -0.7521778 | -1.1050216 | 0.1129203 | 0.4101651 | -1.0422909 | 0.6705582 |
# euclidean distance matrix
dist_eu <- dist(boston_scaled2)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# manhattan distance matrix
dist_man <- dist(boston_scaled2, method = "manhattan")
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
km_boston <- kmeans(boston_scaled2, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km_boston$cluster)
pairs(boston_scaled2[,1:5], col = km_boston$cluster)
pairs(boston_scaled2[,6:10], col = km_boston$cluster)
pairs(boston_scaled2[,11:14], col = km_boston$cluster)
I have performed the k-means clustering with several cluster numbers and think that 3 clusters are the number which allows to get the best overview over the data. The clustering worked good on crim variable, on the nitrogen oxides concentrations and also age, tax and the medv variable. Next we are checking were the k values are decreasing in a plot showing 10 clusters with the total within sum of squares (twcss). On the cluster level where this number decreases heavily that amount of clusters gives a good data overview.
1.3.5. Determine the k
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled2, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km_boston <- kmeans(boston_scaled2, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km_boston$cluster)
The twcss value decrease at a level of 2 - 3 clusters heavily. So the data I think is optimally clustered with three clusters, seen in the overview plot.
1.3.6. Bonus
- Perform the LDA using the clusters as target.
- Include all the variables in the Boston data in the LDA model.
- Visualize the results with a biplot (include arrows representing the relationships of the original variables to the LDA solution)
data(Boston) # load the Boston data set
# scale the Boston data set again - named boston_scaled2
boston_scaled3 <- scale(Boston)
# k-means clustering
km_boston <-kmeans(boston_scaled3, centers = 3)
cluster <- km_boston$cluster
# add the cluster number to the dataframe
boston_scaled3 <- data.frame(boston_scaled3, cluster)
# linear discriminant analysis of clusters vs. all other variables
lda.fit_cluster <- lda(cluster ~ ., data = boston_scaled3)
# print the lda.fit object
lda.fit_cluster## Call:
## lda(cluster ~ ., data = boston_scaled3)
##
## Prior probabilities of groups:
## 1 2 3
## 0.2470356 0.3260870 0.4268775
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3989700 1.2614609 -0.9791535 -0.020354653 -0.8573235 1.0090468
## 2 0.7982270 -0.4872402 1.1186734 0.014005495 1.1351215 -0.4596725
## 3 -0.3788713 -0.3578148 -0.2879024 0.001080671 -0.3709704 -0.2328004
## age dis rad tax ptratio black
## 1 -0.96130713 0.9497716 -0.5867985 -0.6709807 -0.80239137 0.3552363
## 2 0.79930921 -0.8549214 1.2113527 1.2873657 0.59162230 -0.6363367
## 3 -0.05427143 0.1034286 -0.5857564 -0.5951053 0.01241316 0.2805140
## lstat medv
## 1 -0.9571271 1.06668290
## 2 0.8622388 -0.67953738
## 3 -0.1047617 -0.09820229
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim -0.03206338 -0.19094456
## zn 0.02935900 -1.07677218
## indus 0.63347352 -0.09917524
## chas 0.02460719 0.10009606
## nox 1.11749317 -0.75995105
## rm -0.18841682 -0.57360135
## age -0.12983139 0.47226685
## dis 0.04493809 -0.34585958
## rad 0.67004295 -0.08584353
## tax 1.03992455 -0.58075025
## ptratio 0.25864960 -0.02605279
## black -0.01657236 0.01975686
## lstat 0.17365575 -0.41704235
## medv -0.06819126 -0.79098605
##
## Proportion of trace:
## LD1 LD2
## 0.8506 0.1494# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes3 <- as.numeric(boston_scaled3$cluster)# plot the lda results
plot(lda.fit_cluster, dimen = 2, col = classes3, pch = classes3, main = "LDA biplot using three clusters 1, 2 and 3")
lda.arrows(lda.fit_cluster, myscale = 2)
So I scaled the Boston data set. Defined the k-means ‘km_boston’ clusters and added the cluster number to the scaled data frame. Then I ran the LDA model with the clusters against all other variables and added the arrows.
In the LDA biplot you can see how nicely the clusters are separated from each other - so the clustering actually worked. The most influencial linear separators seem to be “age”, “zn” (lands over2500 sq.ft.), “nox” (nitrogen oxide concentration) and “tax” (full-value property-tax rate).
1.3.7. Super-Bonus
#run the code for the scaled training data set
model_predictors <- dplyr::select(boston_train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
# installed plotly package and load it
library(plotly)
crimeplot <- plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = boston_train$crime)
crimeplotn_boston3 <- nrow(boston_scaled3)
# choose randomly 80% of the rows
ind_boston3 <- sample(n_boston3, size = n_boston * 0.8)
# kmeans clustering of boston_scaled3
km_cluster <- kmeans(boston_scaled3, centers = 3)
boston_scaled3$cluster <- km_cluster$cluster
clustertrain <- boston_scaled3[ind_boston3,]
clusterplot <- plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = clustertrain)
clusterplotThe 3D plot of the crime training data set is visible here.
Chapter 5: Dimensionality reduction techniques
Data wrangling and performing principal component analysis (PCA) & multiple correspondence analysis (MCA)
Work of week 48 (25.11. - 01.12.2019)
1. Analysis of human data set
The data wrangling was done in two steps: a data frame combination of two data sets and as a second step a refining the data to those variables we want to analyse. The data wrangling scrip was uploaded to my GitHub repository. You can find the data wrangling script here.
1.1. Load the data set
# load necessary packages
library(tidyr)
library(dplyr)
library(corrplot)
library(ggplot2)
library(GGally)
library(knitr)
library(kableExtra)
library(stringr)
library(ggfortify)
library(factoextra)# load the data set "human"
human <- read.table(file =
"C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/human_analys.txt")1.2. The data set (overview and correlations)
The original complete data set from the United Nations was wrangled and combined into the so called “human” data set. This human data set consists of 8 variables with 155 observations. The data set includes following variables:
- country –> Name of the country
- ratio.sec.edu –> Ratio of second education of female/male
- ratio.lab.force –> Ratio of labour forced female/male
- edu.expect –> Expected years of schooling
- life.exp –> Life expectancy at birth
- GNI –> Gross National Income per capita
- mat.mor.r –> Maternal mortality ratio
- adol.birth –> Adolescent birth rate
- rep.parliament –> Percetange of female representatives in parliament
The data was collected from the United Nations. More information about the data and how it was collected can be found here. Technical notes about calculating the human development indices can be found here.
Here you see the structure, the complete table and the summary of the wrangled human data set.
# check the data set "human"
str(human)## 'data.frame': 155 obs. of 8 variables:
## $ ratio.sec.edu : num 1.007 0.997 0.983 0.989 0.969 ...
## $ ratio.lab.force: num 0.891 0.819 0.825 0.884 0.829 ...
## $ edu.expect : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ life.exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ mat.mor.r : int 4 6 6 5 6 7 9 28 11 8 ...
## $ adol.birth : num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ rep.parliament : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...# data set table
knitr::kable(human) %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>%
scroll_box(width = "100%", height = "300px")| ratio.sec.edu | ratio.lab.force | edu.expect | life.exp | GNI | mat.mor.r | adol.birth | rep.parliament | |
|---|---|---|---|---|---|---|---|---|
| Norway | 1.0072389 | 0.8908297 | 17.5 | 81.6 | 64992 | 4 | 7.8 | 39.6 |
| Australia | 0.9968288 | 0.8189415 | 20.2 | 82.4 | 42261 | 6 | 12.1 | 30.5 |
| Switzerland | 0.9834369 | 0.8251001 | 15.8 | 83.0 | 56431 | 6 | 1.9 | 28.5 |
| Denmark | 0.9886128 | 0.8840361 | 18.7 | 80.2 | 44025 | 5 | 5.1 | 38.0 |
| Netherlands | 0.9690608 | 0.8286119 | 17.9 | 81.6 | 45435 | 6 | 6.2 | 36.9 |
| Germany | 0.9927835 | 0.8072289 | 16.5 | 80.9 | 43919 | 7 | 3.8 | 36.9 |
| Ireland | 1.0241730 | 0.7797357 | 18.6 | 80.9 | 39568 | 9 | 8.2 | 19.9 |
| United States | 1.0031646 | 0.8171263 | 16.5 | 79.1 | 52947 | 28 | 31.0 | 19.4 |
| Canada | 1.0000000 | 0.8676056 | 15.9 | 82.0 | 42155 | 11 | 14.5 | 28.2 |
| New Zealand | 0.9968520 | 0.8401084 | 19.2 | 81.8 | 32689 | 8 | 25.3 | 31.4 |
| Singapore | 0.9148148 | 0.7616580 | 15.4 | 83.0 | 76628 | 6 | 6.0 | 25.3 |
| Sweden | 0.9908362 | 0.8880707 | 15.8 | 82.2 | 45636 | 4 | 6.5 | 43.6 |
| United Kingdom | 0.9989990 | 0.8107715 | 16.2 | 80.7 | 39267 | 8 | 25.8 | 23.5 |
| Iceland | 0.9934498 | 0.9108527 | 19.0 | 82.6 | 35182 | 4 | 11.5 | 41.3 |
| Korea (Republic of) | 0.8641975 | 0.6948682 | 16.9 | 81.9 | 33890 | 27 | 2.2 | 16.3 |
| Israel | 0.9667812 | 0.8379161 | 16.0 | 82.4 | 30676 | 2 | 7.8 | 22.5 |
| Luxembourg | 1.0000000 | 0.7848297 | 13.9 | 81.7 | 58711 | 11 | 8.3 | 28.3 |
| Japan | 1.0139860 | 0.6931818 | 15.3 | 83.5 | 36927 | 6 | 5.4 | 11.6 |
| Belgium | 0.9348613 | 0.8010118 | 16.3 | 80.8 | 41187 | 6 | 6.7 | 42.4 |
| France | 0.9375000 | 0.8230519 | 16.0 | 82.2 | 38056 | 12 | 5.7 | 25.7 |
| Austria | 1.0000000 | 0.8064993 | 15.7 | 81.4 | 43869 | 4 | 4.1 | 30.3 |
| Finland | 1.0000000 | 0.8703125 | 17.1 | 80.8 | 38695 | 4 | 9.2 | 42.5 |
| Slovenia | 0.9775510 | 0.8275316 | 16.8 | 80.4 | 27852 | 7 | 0.6 | 27.7 |
| Spain | 0.9138167 | 0.7978723 | 17.3 | 82.6 | 32045 | 4 | 10.6 | 38.0 |
| Italy | 0.8844720 | 0.6655462 | 16.0 | 83.1 | 33030 | 4 | 4.0 | 30.1 |
| Czech Republic | 1.0020060 | 0.7481698 | 16.4 | 78.6 | 26660 | 5 | 4.9 | 18.9 |
| Greece | 0.8880597 | 0.7072000 | 17.6 | 80.9 | 24524 | 5 | 11.9 | 21.0 |
| Estonia | 1.0000000 | 0.8156749 | 16.5 | 76.8 | 25214 | 11 | 16.8 | 19.8 |
| Cyprus | 0.9302326 | 0.7876231 | 14.0 | 80.2 | 28633 | 10 | 5.5 | 12.5 |
| Qatar | 1.1305085 | 0.5319372 | 13.8 | 78.2 | 123124 | 6 | 9.5 | 0.0 |
| Slovakia | 0.9959799 | 0.7448980 | 15.1 | 76.3 | 25845 | 7 | 15.9 | 18.7 |
| Poland | 0.9286550 | 0.7534669 | 15.5 | 77.4 | 23177 | 3 | 12.2 | 22.1 |
| Lithuania | 0.9448568 | 0.8291233 | 16.4 | 73.3 | 24500 | 11 | 10.6 | 23.4 |
| Malta | 0.8772379 | 0.5716440 | 14.4 | 80.6 | 27930 | 9 | 18.2 | 13.0 |
| Saudi Arabia | 0.8605974 | 0.2579821 | 16.3 | 74.3 | 52821 | 16 | 10.2 | 19.9 |
| Argentina | 0.9774306 | 0.6333333 | 17.9 | 76.3 | 22050 | 69 | 54.4 | 36.8 |
| United Arab Emirates | 1.1944444 | 0.5054348 | 13.3 | 77.0 | 60868 | 8 | 27.6 | 17.5 |
| Chile | 0.9594241 | 0.6577540 | 15.2 | 81.7 | 21290 | 22 | 55.3 | 15.8 |
| Portugal | 0.9896266 | 0.8293051 | 16.3 | 80.9 | 25757 | 8 | 12.6 | 31.3 |
| Hungary | 0.9918946 | 0.7466667 | 15.4 | 75.2 | 22916 | 14 | 12.1 | 10.1 |
| Bahrain | 1.1031128 | 0.4510932 | 14.4 | 76.6 | 38599 | 22 | 13.8 | 15.0 |
| Latvia | 0.9989899 | 0.8121302 | 15.2 | 74.2 | 22281 | 13 | 13.5 | 18.0 |
| Croatia | 0.9081197 | 0.7654110 | 14.8 | 77.3 | 19409 | 13 | 12.7 | 25.8 |
| Kuwait | 0.9875666 | 0.5246691 | 14.7 | 74.4 | 83961 | 14 | 14.5 | 1.5 |
| Montenegro | 0.8891235 | 0.7504363 | 15.2 | 76.2 | 14558 | 7 | 15.2 | 17.3 |
| Belarus | 0.9436009 | 0.7939778 | 15.7 | 71.3 | 16676 | 1 | 20.6 | 30.1 |
| Russian Federation | 0.9686486 | 0.7963738 | 14.7 | 70.1 | 22352 | 24 | 25.7 | 14.5 |
| Oman | 0.8266200 | 0.3510896 | 13.6 | 76.8 | 34858 | 11 | 10.6 | 9.6 |
| Romania | 0.9358696 | 0.7503852 | 14.2 | 74.7 | 18108 | 33 | 31.0 | 12.0 |
| Uruguay | 1.0815109 | 0.7239583 | 15.5 | 77.2 | 19283 | 14 | 58.3 | 11.5 |
| Bahamas | 1.0410959 | 0.8738966 | 12.6 | 75.4 | 21336 | 37 | 28.5 | 16.7 |
| Kazakhstan | 0.9645749 | 0.8690629 | 15.0 | 69.4 | 20867 | 26 | 29.9 | 20.1 |
| Barbados | 1.0205245 | 0.8603133 | 15.4 | 75.6 | 12488 | 52 | 48.4 | 19.6 |
| Bulgaria | 0.9717868 | 0.8118644 | 14.4 | 74.2 | 15596 | 5 | 35.9 | 20.4 |
| Panama | 1.0821643 | 0.5990220 | 13.3 | 77.6 | 18192 | 85 | 78.5 | 19.3 |
| Malaysia | 0.9130435 | 0.5880795 | 12.7 | 74.7 | 22762 | 29 | 5.7 | 14.2 |
| Mauritius | 0.8517241 | 0.5876011 | 15.6 | 74.4 | 17470 | 73 | 30.9 | 11.6 |
| Trinidad and Tobago | 0.9802956 | 0.7019868 | 12.3 | 70.4 | 26090 | 84 | 34.8 | 24.7 |
| Serbia | 0.7934783 | 0.7307061 | 14.4 | 74.9 | 12190 | 16 | 16.9 | 34.0 |
| Cuba | 0.9428934 | 0.6200000 | 13.8 | 79.4 | 7301 | 80 | 43.1 | 48.9 |
| Lebanon | 0.9566787 | 0.3286319 | 13.8 | 79.3 | 16509 | 16 | 12.0 | 3.1 |
| Costa Rica | 1.0039604 | 0.5898734 | 13.9 | 79.4 | 13413 | 38 | 60.8 | 33.3 |
| Iran (Islamic Republic of) | 0.9201183 | 0.2255435 | 15.1 | 75.4 | 15440 | 23 | 31.6 | 3.1 |
| Venezuela (Bolivarian Republic of) | 1.1141732 | 0.6452020 | 14.2 | 74.2 | 16159 | 110 | 83.2 | 17.0 |
| Turkey | 0.6500000 | 0.4152542 | 14.5 | 75.3 | 18677 | 20 | 30.9 | 14.4 |
| Sri Lanka | 0.9515707 | 0.4600262 | 13.7 | 74.9 | 9779 | 29 | 16.9 | 5.8 |
| Mexico | 0.9191419 | 0.5644556 | 13.1 | 76.8 | 16056 | 49 | 63.4 | 37.1 |
| Brazil | 1.0419847 | 0.7351485 | 15.2 | 74.5 | 15175 | 69 | 70.8 | 9.6 |
| Georgia | 0.9676375 | 0.7523302 | 13.8 | 74.9 | 7164 | 41 | 46.8 | 11.3 |
| Azerbaijan | 0.9620123 | 0.9037356 | 11.9 | 70.8 | 16428 | 26 | 40.0 | 15.6 |
| Jordan | 0.8853503 | 0.2342342 | 13.5 | 74.0 | 11365 | 50 | 26.5 | 11.6 |
| The former Yugoslav Republic of Macedonia | 0.7230216 | 0.6385185 | 13.4 | 75.4 | 11780 | 7 | 18.3 | 33.3 |
| Ukraine | 0.9562044 | 0.7952167 | 15.1 | 71.0 | 8178 | 23 | 25.7 | 11.8 |
| Algeria | 0.8612903 | 0.2105263 | 14.0 | 74.8 | 13054 | 89 | 10.0 | 25.7 |
| Peru | 0.8517398 | 0.8080569 | 13.1 | 74.6 | 11015 | 89 | 50.7 | 22.3 |
| Albania | 0.9306030 | 0.6854962 | 11.8 | 77.8 | 9943 | 21 | 15.3 | 20.7 |
| Armenia | 0.9894737 | 0.7465565 | 12.3 | 74.7 | 8124 | 29 | 27.1 | 10.7 |
| Bosnia and Herzegovina | 0.6432665 | 0.5951134 | 13.6 | 76.5 | 9638 | 8 | 15.1 | 19.3 |
| Ecuador | 1.0177665 | 0.6614268 | 14.2 | 75.9 | 10605 | 87 | 77.0 | 41.6 |
| China | 0.8164117 | 0.8160920 | 13.1 | 75.8 | 12547 | 32 | 8.6 | 23.6 |
| Fiji | 0.9953488 | 0.5208333 | 15.7 | 70.0 | 7493 | 59 | 42.8 | 14.0 |
| Mongolia | 1.0142687 | 0.8167388 | 14.6 | 69.4 | 10729 | 68 | 18.7 | 14.9 |
| Thailand | 0.8750000 | 0.7967782 | 13.5 | 74.4 | 13323 | 26 | 41.0 | 6.1 |
| Libya | 1.3245823 | 0.3926702 | 14.0 | 71.6 | 14911 | 15 | 2.5 | 16.0 |
| Tunisia | 0.7114967 | 0.3540197 | 14.6 | 74.8 | 10404 | 46 | 4.6 | 31.3 |
| Colombia | 1.0233813 | 0.7001255 | 13.5 | 74.0 | 12040 | 83 | 68.5 | 20.9 |
| Jamaica | 1.0541311 | 0.7912553 | 12.4 | 75.7 | 7415 | 80 | 70.1 | 16.7 |
| Tonga | 0.9909400 | 0.7171582 | 14.7 | 72.8 | 5069 | 120 | 18.1 | 0.0 |
| Belize | 1.0079156 | 0.5978129 | 13.6 | 70.0 | 7614 | 45 | 71.4 | 13.3 |
| Dominican Republic | 1.0470810 | 0.6526718 | 13.1 | 73.5 | 11883 | 100 | 99.6 | 19.1 |
| Suriname | 0.9469214 | 0.5886628 | 12.7 | 71.1 | 15617 | 130 | 35.2 | 11.8 |
| Maldives | 0.8348624 | 0.7251613 | 13.0 | 76.8 | 12328 | 31 | 4.2 | 5.9 |
| Samoa | 1.0716667 | 0.4023973 | 12.9 | 73.4 | 5327 | 58 | 28.3 | 6.1 |
| Botswana | 0.9448010 | 0.8811275 | 12.5 | 64.5 | 16646 | 170 | 44.2 | 9.5 |
| Moldova (Republic of) | 0.9689441 | 0.8506787 | 11.9 | 71.6 | 5223 | 21 | 29.3 | 20.8 |
| Egypt | 0.7244224 | 0.3168449 | 13.5 | 71.1 | 10512 | 45 | 43.0 | 2.2 |
| Gabon | 1.4930748 | 0.8593272 | 12.5 | 64.4 | 16367 | 240 | 103.0 | 16.2 |
| Indonesia | 0.8109756 | 0.6104513 | 13.0 | 68.9 | 9788 | 190 | 48.3 | 17.1 |
| Paraguay | 0.8558140 | 0.6568396 | 11.9 | 72.9 | 7643 | 110 | 67.0 | 16.8 |
| Philippines | 1.0345369 | 0.6411543 | 11.3 | 68.2 | 7915 | 120 | 46.8 | 27.1 |
| El Salvador | 0.8440367 | 0.6050633 | 12.3 | 73.0 | 7349 | 69 | 76.0 | 27.4 |
| South Africa | 0.9578393 | 0.7355372 | 13.6 | 57.4 | 12122 | 140 | 50.9 | 40.7 |
| Viet Nam | 0.8342697 | 0.8880779 | 11.9 | 75.8 | 5092 | 49 | 29.0 | 24.3 |
| Bolivia (Plurinational State of) | 0.8054146 | 0.7935723 | 13.2 | 68.3 | 5760 | 200 | 71.9 | 51.8 |
| Kyrgyzstan | 0.9762397 | 0.7044025 | 12.5 | 70.6 | 3044 | 75 | 29.3 | 23.3 |
| Iraq | 0.5537849 | 0.2134670 | 10.1 | 69.4 | 14003 | 67 | 68.7 | 26.5 |
| Guyana | 1.2615063 | 0.5291925 | 10.3 | 66.4 | 6522 | 250 | 88.5 | 31.3 |
| Nicaragua | 1.0287206 | 0.5902864 | 11.5 | 74.9 | 4457 | 100 | 100.8 | 39.1 |
| Morocco | 0.6854305 | 0.3496042 | 11.6 | 74.0 | 6850 | 120 | 35.8 | 11.0 |
| Namibia | 0.9680233 | 0.8587127 | 11.3 | 64.8 | 9418 | 130 | 54.9 | 37.7 |
| Guatemala | 0.9439655 | 0.5589569 | 10.7 | 71.8 | 6929 | 140 | 97.2 | 13.3 |
| Tajikistan | 1.0427632 | 0.7639429 | 11.2 | 69.4 | 2517 | 44 | 42.8 | 15.2 |
| India | 0.4770318 | 0.3379224 | 11.7 | 68.0 | 5497 | 190 | 32.8 | 12.2 |
| Honduras | 1.0852713 | 0.5162847 | 11.1 | 73.1 | 3938 | 120 | 84.0 | 25.8 |
| Bhutan | 0.9855072 | 0.8639896 | 12.6 | 69.5 | 7176 | 120 | 40.9 | 8.3 |
| Syrian Arab Republic | 0.7283951 | 0.1856946 | 12.3 | 69.6 | 2728 | 49 | 41.6 | 12.4 |
| Congo | 0.8446809 | 0.9383562 | 11.1 | 62.3 | 6012 | 410 | 126.7 | 11.5 |
| Zambia | 0.5863636 | 0.8539720 | 13.5 | 60.1 | 3734 | 280 | 125.4 | 12.7 |
| Ghana | 0.6986090 | 0.9425770 | 11.5 | 61.4 | 3852 | 380 | 58.4 | 10.9 |
| Bangladesh | 0.8256659 | 0.6825208 | 10.0 | 71.6 | 3191 | 170 | 80.6 | 20.0 |
| Cambodia | 0.4323144 | 0.9109827 | 10.9 | 68.4 | 2949 | 170 | 44.3 | 19.0 |
| Kenya | 0.8057325 | 0.8591160 | 11.0 | 61.6 | 2762 | 400 | 93.6 | 20.8 |
| Nepal | 0.4633508 | 0.9173364 | 12.4 | 69.6 | 2311 | 190 | 73.7 | 29.5 |
| Pakistan | 0.4186551 | 0.2967431 | 7.8 | 66.2 | 4866 | 170 | 27.3 | 19.7 |
| Myanmar | 1.4967320 | 0.9137303 | 8.6 | 65.9 | 4608 | 200 | 12.1 | 4.7 |
| Swaziland | 0.8423077 | 0.6131285 | 11.3 | 49.0 | 5542 | 310 | 72.0 | 14.7 |
| Tanzania (United Republic of) | 0.5894737 | 0.9767184 | 9.2 | 65.0 | 2411 | 410 | 122.7 | 36.0 |
| Cameroon | 0.6103152 | 0.8307292 | 10.4 | 55.5 | 2803 | 590 | 115.8 | 27.1 |
| Zimbabwe | 0.7854839 | 0.9275362 | 10.9 | 57.5 | 1615 | 470 | 60.3 | 35.1 |
| Mauritania | 0.3971292 | 0.3628319 | 8.5 | 63.1 | 3560 | 320 | 73.3 | 22.2 |
| Papua New Guinea | 0.5241379 | 0.9527027 | 9.9 | 62.6 | 2463 | 220 | 62.1 | 2.7 |
| Yemen | 0.3220974 | 0.3518006 | 9.2 | 63.8 | 3519 | 270 | 47.0 | 0.7 |
| Lesotho | 1.1526316 | 0.8027211 | 11.1 | 49.8 | 3306 | 490 | 89.4 | 26.8 |
| Togo | 0.3995037 | 0.9913899 | 12.2 | 59.7 | 1228 | 450 | 91.5 | 17.6 |
| Haiti | 0.6363636 | 0.8577465 | 8.7 | 62.8 | 1669 | 380 | 42.0 | 3.5 |
| Rwanda | 0.9090909 | 1.0128957 | 10.3 | 64.2 | 1458 | 320 | 33.6 | 57.5 |
| Uganda | 0.6835821 | 0.9570707 | 9.8 | 58.5 | 1613 | 360 | 126.6 | 35.0 |
| Benin | 0.4185185 | 0.8633461 | 11.1 | 59.6 | 1767 | 340 | 90.2 | 8.4 |
| Sudan | 0.6648352 | 0.4118421 | 7.0 | 63.5 | 3809 | 360 | 84.0 | 23.8 |
| Senegal | 0.4675325 | 0.7500000 | 7.9 | 66.5 | 2188 | 320 | 94.4 | 42.7 |
| Afghanistan | 0.1979866 | 0.1987421 | 9.3 | 60.4 | 1885 | 400 | 86.8 | 27.6 |
| Côte d’Ivoire | 0.4651163 | 0.6437346 | 8.9 | 51.5 | 3171 | 720 | 130.3 | 9.2 |
| Malawi | 0.5138889 | 1.0380368 | 10.8 | 62.8 | 747 | 510 | 144.8 | 16.7 |
| Ethiopia | 0.4285714 | 0.8756999 | 8.5 | 64.1 | 1428 | 420 | 78.4 | 25.5 |
| Gambia | 0.5523810 | 0.8709288 | 8.8 | 60.2 | 1507 | 430 | 115.8 | 9.4 |
| Congo (Democratic Republic of the) | 0.3950617 | 0.9658470 | 9.8 | 58.7 | 680 | 730 | 135.3 | 8.2 |
| Liberia | 0.3918575 | 0.8981481 | 9.5 | 60.9 | 805 | 640 | 117.4 | 10.7 |
| Mali | 0.5099338 | 0.6240786 | 8.4 | 58.0 | 1583 | 550 | 175.6 | 9.5 |
| Mozambique | 0.2258065 | 1.0326087 | 9.3 | 55.1 | 1123 | 480 | 137.8 | 39.6 |
| Sierra Leone | 0.4608295 | 0.9521739 | 8.6 | 50.9 | 1780 | 1100 | 100.7 | 12.4 |
| Burkina Faso | 0.2812500 | 0.8566667 | 7.8 | 58.7 | 1591 | 400 | 115.4 | 13.3 |
| Burundi | 0.6385542 | 1.0158537 | 10.1 | 56.7 | 758 | 740 | 30.3 | 34.9 |
| Chad | 0.1717172 | 0.8080808 | 7.4 | 51.6 | 2085 | 980 | 152.0 | 14.9 |
| Central African Republic | 0.3782772 | 0.8531140 | 7.2 | 50.7 | 581 | 880 | 98.3 | 12.5 |
| Niger | 0.3076923 | 0.4459309 | 5.4 | 61.4 | 908 | 630 | 204.8 | 13.3 |
# data summary
knitr::kable(summary(human)) %>%
kable_styling(bootstrap_options = "striped", position = "center", font_size = 11)| ratio.sec.edu | ratio.lab.force | edu.expect | life.exp | GNI | mat.mor.r | adol.birth | rep.parliament | |
|---|---|---|---|---|---|---|---|---|
| Min. :0.1717 | Min. :0.1857 | Min. : 5.40 | Min. :49.00 | Min. : 581 | Min. : 1.0 | Min. : 0.60 | Min. : 0.00 | |
| 1st Qu.:0.7264 | 1st Qu.:0.5984 | 1st Qu.:11.25 | 1st Qu.:66.30 | 1st Qu.: 4198 | 1st Qu.: 11.5 | 1st Qu.: 12.65 | 1st Qu.:12.40 | |
| Median :0.9375 | Median :0.7535 | Median :13.50 | Median :74.20 | Median : 12040 | Median : 49.0 | Median : 33.60 | Median :19.30 | |
| Mean :0.8529 | Mean :0.7074 | Mean :13.18 | Mean :71.65 | Mean : 17628 | Mean : 149.1 | Mean : 47.16 | Mean :20.91 | |
| 3rd Qu.:0.9968 | 3rd Qu.:0.8535 | 3rd Qu.:15.20 | 3rd Qu.:77.25 | 3rd Qu.: 24512 | 3rd Qu.: 190.0 | 3rd Qu.: 71.95 | 3rd Qu.:27.95 | |
| Max. :1.4967 | Max. :1.0380 | Max. :20.20 | Max. :83.50 | Max. :123124 | Max. :1100.0 | Max. :204.80 | Max. :57.50 |
1.2.1. Graphical overview of the human data set
# visualization of human data set
ov_human <- ggpairs(human, mapping = aes(), title ="Overview of the human data set",
lower = list(combo = wrap("facethist", bins = 20)),
upper = list(continuous = wrap("cor", size = 3)))
ov_human
The overview plot shows the data distributions of all variables in the data set and its correlations to each other. Most of the data is basically normally distributed. GNI, maternal mortality rate and adolescence birth rate show left skewed distribution representing a increased cound on low rates. On the upper part the correlations of the variables are shown as number values. Following you see the correlation matrix and a correlation plot presenting the values graphically.
# calculate the correlation matrix and round it
cor_human <- cor(human) %>% round(digits = 2)
cor_human %>% knitr::kable(caption = "Correlation table of human data set") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| ratio.sec.edu | ratio.lab.force | edu.expect | life.exp | GNI | mat.mor.r | adol.birth | rep.parliament | |
|---|---|---|---|---|---|---|---|---|
| ratio.sec.edu | 1.00 | 0.01 | 0.59 | 0.58 | 0.43 | -0.66 | -0.53 | 0.08 |
| ratio.lab.force | 0.01 | 1.00 | 0.05 | -0.14 | -0.02 | 0.24 | 0.12 | 0.25 |
| edu.expect | 0.59 | 0.05 | 1.00 | 0.79 | 0.62 | -0.74 | -0.70 | 0.21 |
| life.exp | 0.58 | -0.14 | 0.79 | 1.00 | 0.63 | -0.86 | -0.73 | 0.17 |
| GNI | 0.43 | -0.02 | 0.62 | 0.63 | 1.00 | -0.50 | -0.56 | 0.09 |
| mat.mor.r | -0.66 | 0.24 | -0.74 | -0.86 | -0.50 | 1.00 | 0.76 | -0.09 |
| adol.birth | -0.53 | 0.12 | -0.70 | -0.73 | -0.56 | 0.76 | 1.00 | -0.07 |
| rep.parliament | 0.08 | 0.25 | 0.21 | 0.17 | 0.09 | -0.09 | -0.07 | 1.00 |
# Specialized the insignificant value according to the significant level
p.mat <- cor.mtest(cor_human)$p
# visualize the correlation matrix
# correlations / colour shows the correlation values
corrplot(cor_human, method="pie", type="lower", tl.cex = 0.65, p.mat = p.mat, sig.level = 0.01, tl.srt = 45, title="Correlations of the human data set", mar=c(0,0,1,0)) 
This correlation plot shows nicely which variable corrate positively or negatively to each other. The crossed square show a correlation which is not significant (p-value > 0.01). Strong positive correlations are seen between live expectation and expected years of education, gross national income with expected years of education and live expectation and adolescent birth rate with maternal mortality. So a better education leads to a higher income and so a higher life expectation. Maternal mortality rate depends on the rate of adolescence birth rate.
1.3. Data Analysis
1.3.1. Principal component analysis (PCA) on the not standardized human data set
Generally: in a PCA the data is transformed into new features called the principal components. The first principal component (PC) captures the maximum amount of variance of the features in the original data. The second PC captures the maximum amount of variability left and is orthogonal to the first PC (in a right angle to the first PC). All PCAs are uncorrelated analysis methods. Here we are using PC1 and PC2 for the data interpretation.
Here the principal component analysis of the not standardized human data set, a table with the values of the principle components 1 & 2 and the PCA biplot.
# perform PCA on not standardized data set
pca_human <- prcomp(human)
# PCA results on PC1 and PC2 on all variables
round(pca_human$rotation[,1:2], digits = 4) %>% knitr::kable(caption = "PCA result on all variables") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| PC1 | PC2 | |
|---|---|---|
| ratio.sec.edu | 0.0000 | 0.0007 |
| ratio.lab.force | 0.0000 | -0.0003 |
| edu.expect | -0.0001 | 0.0076 |
| life.exp | -0.0003 | 0.0283 |
| GNI | -1.0000 | -0.0058 |
| mat.mor.r | 0.0057 | -0.9916 |
| adol.birth | 0.0012 | -0.1256 |
| rep.parliament | -0.0001 | 0.0032 |
# summary of the pca
s1 <- summary(pca_human)
# round the percentages of variance captured by the pc
pca_prc1 <- round(100 * s1$importance[2, ], digits = 3)
# Principal components 1 & 2 table
round(100 * s1$importance[2, 1:2], digits = 5) %>% knitr::kable(caption = "Principle components") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| x | |
|---|---|
| PC1 | 99.99 |
| PC2 | 0.01 |
# prepare a label for the biplot
pca_label1 <- paste0("GNI (", pca_prc1[1],"%)" )
pca_label2 <- paste0("Maternal mortality (", pca_prc1[2],"%)" )
# draw the pca biplot
biplot(pca_human, choices = 1:2, cex = c(0.55, 0,9), col = c("grey40", "darkgreen"), xlab = pca_label1,
ylab = pca_label2, main = "PCA biplot of non standardized human data set", margin =c(0,0,5,0)) 
Let’s try another kind of biplot
# more advanced biplot
autoplot(pca_human, data = human, label= TRUE, label.size = 3.0, colour = "darkgreen", loadings = TRUE, loadings.label = TRUE, loadings.colour = "red",) + ggtitle("PCA biplot of not standardized human data set") + xlab(paste0(pca_label1)) + ylab(paste0(pca_label2)) + theme_bw()
Principal component 1 represents 99.99% of the variance of the data set. PC2 captures just 0.01% of the variance. Pc1 is associated with the GNI variable. So, the Gross National Income per capita explains 99.99% of the principal component 1 of the data. 0.01 % is associated with maternal mortalility rate. This is the not standardized data set, that’s why we get this result.
1.3.2. Principal component analysis (PCA) on the standardized human data set
First we are scaling the data set so that the human is standardized so that the mean of all values is 0 and the standard diviation reaches 1. See here the summary table of the standardized data set.
# scaling of human data set
human_stzd <- scale(human)
# summary of standardized human data set
knitr::kable(summary(human_stzd)) %>%
kable_styling(bootstrap_options = "striped", position = "center", font_size = 11)| ratio.sec.edu | ratio.lab.force | edu.expect | life.exp | GNI | mat.mor.r | adol.birth | rep.parliament | |
|---|---|---|---|---|---|---|---|---|
| Min. :-2.8189 | Min. :-2.6247 | Min. :-2.7378 | Min. :-2.7188 | Min. :-0.9193 | Min. :-0.6992 | Min. :-1.1325 | Min. :-1.8203 | |
| 1st Qu.:-0.5233 | 1st Qu.:-0.5484 | 1st Qu.:-0.6782 | 1st Qu.:-0.6425 | 1st Qu.:-0.7243 | 1st Qu.:-0.6496 | 1st Qu.:-0.8394 | 1st Qu.:-0.7409 | |
| Median : 0.3503 | Median : 0.2316 | Median : 0.1140 | Median : 0.3056 | Median :-0.3013 | Median :-0.4726 | Median :-0.3298 | Median :-0.1403 | |
| Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | Mean : 0.0000 | |
| 3rd Qu.: 0.5958 | 3rd Qu.: 0.7350 | 3rd Qu.: 0.7126 | 3rd Qu.: 0.6717 | 3rd Qu.: 0.3712 | 3rd Qu.: 0.1932 | 3rd Qu.: 0.6030 | 3rd Qu.: 0.6127 | |
| Max. : 2.6646 | Max. : 1.6632 | Max. : 2.4730 | Max. : 1.4218 | Max. : 5.6890 | Max. : 4.4899 | Max. : 3.8344 | Max. : 3.1850 |
In the summary table you can see that all mean values are 0. So the scaling of the human data set was successful.
Now the PCA of this standardized data follows.
# perform PCA on standardized data set
pca_human_stzd <- prcomp(human_stzd)
# Principal components 1 & 2 table
round(pca_human_stzd$rotation[, 1:2], digits = 4) %>% knitr::kable(caption = "PC1 & 2 of standardized human data") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| PC1 | PC2 | |
|---|---|---|
| ratio.sec.edu | -0.3566 | 0.0380 |
| ratio.lab.force | 0.0546 | 0.7243 |
| edu.expect | -0.4277 | 0.1394 |
| life.exp | -0.4437 | -0.0253 |
| GNI | -0.3505 | 0.0506 |
| mat.mor.r | 0.4370 | 0.1451 |
| adol.birth | 0.4113 | 0.0771 |
| rep.parliament | -0.0844 | 0.6514 |
# PC labels
s2 <- summary(pca_human_stzd)
pca_prc2 <- round(100 * s2$importance[2, ], digits = 1)
pca_label2.1 <- paste0("Education and health (",pca_prc2[1],"%) ")
pca_label2.2 <- paste0("Female social participation (",pca_prc2[2],"%) ")
# Principal components 1 & 2 table
round(100 * s2$importance[2, 1:2], digits = 2) %>% knitr::kable(caption = "Principle components") %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| x | |
|---|---|
| PC1 | 53.61 |
| PC2 | 16.24 |
# draw the PCA biplot
biplot(pca_human_stzd, choices = 1:2, cex = c(0.5, 0,9), col = c("grey40", "deeppink2"), xlab = pca_label2.1, ylab = pca_label2.2, main = "PCA biplot of standardized human data set", margin =c(0,0,5,0))
Also here another kind of biplot
# more advanced biplot
autoplot(prcomp(human_stzd), data = human, colour = "darkgreen", label= TRUE, label.size = 3.0, loadings = TRUE, loadings.label = TRUE, loadings.colour = "red",) + ggtitle("PCA biplot of standardized human data set") + xlab(paste0(pca_label2.1)) + ylab(paste0(pca_label2.2)) + theme_bw()
The principal component representing 16.2% of the data variation is strongly associated with parliament representatives and the labour force ratio (female / male). So the occurance of of woman in the labour market and in the political representation is mostly associated with principal component 2. The variables which show the strongest association with principal component 1 (representing 53.6% of the data variation) are life expectation, the school period, the ratio of female/male of second education, adolenscence birthrate and maternal mortality. So overall principal component 1 represents the health and education situation of the countries.
The PCA results differ a lot! To perform the PCA we need to make sure that the obsvervations are standardized, which means that the values are transformed in compareable values.
We can see that in the PCA of the non-standardized human data set the principal component 1 represents almost all varation of the data set - so the Gross National Income per capita is the value representing principle component 1.
The PCA of standardized values looks quite different. You can see that the PC1 & PC2 values spread all over the biplot and more variables are associated to the two principal components.
The angle between the arrows represents the original features of the data set and we can interpret it as the correlation between these features. Is the angle between the arrows small, then there is a positive correlation.
Also the angle between the feature and the principal component can be interpreted as the correlation between these two components. Again, if the angle is small there is a positive correlation. The length of the arrows is proportional to the standard deviation of the features.
2. Multiple Correspondence Analysis (MCA)
2.1. Tea consumption behaviour MCA
The “tea” data set includes data about tea consumption. Following, you see the MCA of the tea data set as it was performed on the DataCamp exercise.
2.1.1. Load the data set & check it
# load the package and the tea data set
library(FactoMineR)
data(tea)
# check the tea data set
str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...2.1.2. Data overview
# column names to keep in the dataset
keep_tea <- c("Tea", "How", "how", "sugar", "where", "lunch") # from DataCamp
# select the 'keep_columns' to create a new dataset
tea_time <- select(tea, one_of(keep_tea))
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar(fill = "darkgreen") + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
2.1.3. MCA
# multiple correspondence analysis
mca_tea_time <- MCA(tea_time, graph = FALSE)
# summary of the model
summary(mca_tea_time)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |# visualize MCA
plot(mca_tea_time, invisible=c("ind"), habillage = "quali")
2.1.4. MCA, different biplot using “factoextra” package
Here the MCA presented in another MCA biplot, performed with the function ‘fviz_mca_biplot()’ which is available from the ‘factoextra’ package. Here I added as a individual value colour the cosine-squared (cos2), a value between 0 & 1. As closer to 1 this value gets as better it is projected on the dimension.
#draw a MCA biplot using factoextra function "fviz_mca_biplot"
fviz_mca_biplot(mca_tea_time, col.ind = "cos2", col.var = "red", label = "var", geom = c("point","text"), labelsize = 4, arrows = c(FALSE,TRUE)) + labs(title = "MCA of tea consumption behaviour") + theme_grey() + theme(axis.line = element_line(size = 0.5), panel.background = element_rect(fill = "gray93"))
2.2. Another MCA analysis of different “tea” variables (reasons for drinking tea)
I want to know which reasons for drinking tea people have. So I have choosen following six variables to analyse by MCA:
- sex –> Gender of the tea consumer
- age_Q –> Age class of the tea consumer
- spirituality –> Spiritual reason to drink tea
- healthy –> Health reason to drink tea
- slimming –> Tea consumer wants to loose weight or not
- relaxing –> Relaxation effect on tea on consumer or not
I prepare a separate tea_reason data set and perform the MCA of that data aliquote.
2.2.1. Prepare the data set and overview graph
# column names to keep for my analysis
keep_reason <- c("sex", "age_Q", "spirituality", "healthy", "slimming", "relaxing")
# select the 'keep_columns' to create a new dataset
tea_reason <- select(tea, one_of(keep_reason))
gather(tea_reason) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar(fill = "darkviolet", alpha = 0.6) + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
2.2.2. MCA of the “tea_reason” data
# multiple correspondence analysis
mca_tea_reason <- MCA(tea_reason, graph = FALSE)
# summary of the model
summary(mca_tea_reason)##
## Call:
## MCA(X = tea_reason, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.233 0.217 0.202 0.173 0.166 0.138
## % of var. 15.562 14.476 13.456 11.502 11.066 9.187
## Cumulative % of var. 15.562 30.039 43.495 54.997 66.063 75.251
## Dim.7 Dim.8 Dim.9
## Variance 0.132 0.126 0.113
## % of var. 8.833 8.393 7.523
## Cumulative % of var. 84.083 92.477 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | 0.491 0.344 0.135 | 0.199 0.061 0.022 | 0.110
## 2 | 0.506 0.366 0.210 | 0.599 0.551 0.294 | -0.564
## 3 | 0.039 0.002 0.001 | 0.353 0.191 0.119 | -0.380
## 4 | -0.602 0.518 0.306 | -0.531 0.432 0.237 | 0.178
## 5 | 0.208 0.062 0.024 | -0.446 0.305 0.112 | -0.116
## 6 | -0.270 0.104 0.081 | -0.288 0.128 0.093 | 0.198
## 7 | 0.024 0.001 0.000 | -0.047 0.003 0.001 | 0.294
## 8 | -0.193 0.053 0.025 | 0.146 0.033 0.014 | -0.211
## 9 | 0.308 0.135 0.049 | -0.410 0.259 0.088 | 0.073
## 10 | 0.024 0.001 0.000 | -0.047 0.003 0.001 | 0.294
## ctr cos2
## 1 0.020 0.007 |
## 2 0.526 0.261 |
## 3 0.239 0.138 |
## 4 0.052 0.027 |
## 5 0.022 0.008 |
## 6 0.065 0.044 |
## 7 0.143 0.054 |
## 8 0.074 0.030 |
## 9 0.009 0.003 |
## 10 0.143 0.054 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## F | -0.256 2.767 0.095 -5.338 | 0.220 2.198
## M | 0.373 4.037 0.095 5.338 | -0.321 3.208
## 15-24 | -0.834 15.237 0.308 -9.593 | -0.524 6.472
## 25-34 | 0.681 7.619 0.139 6.437 | -0.744 9.760
## 35-44 | 0.018 0.003 0.000 0.121 | 0.149 0.229
## 45-59 | 0.692 6.948 0.122 6.043 | 0.728 8.276
## +60 | -0.346 1.086 0.017 -2.282 | 1.293 16.262
## Not.spirituality | 0.302 4.458 0.199 7.719 | 0.212 2.374
## spirituality | -0.661 9.770 0.199 -7.719 | -0.465 5.203
## healthy | -0.247 3.050 0.142 -6.525 | 0.304 4.981
## cos2 v.test Dim.3 ctr cos2 v.test
## F 0.070 4.589 | -0.555 15.066 0.449 -11.582 |
## M 0.070 -4.589 | 0.809 21.981 0.449 11.582 |
## 15-24 0.122 -6.030 | -0.522 6.893 0.120 -6.000 |
## 25-34 0.165 -7.027 | 0.983 18.367 0.289 9.294 |
## 35-44 0.003 1.014 | -0.261 0.750 0.010 -1.770 |
## 45-59 0.135 6.361 | -0.716 8.605 0.131 -6.254 |
## +60 0.243 8.517 | 0.901 8.498 0.118 5.936 |
## Not.spirituality 0.099 5.433 | 0.017 0.016 0.001 0.432 |
## spirituality 0.099 -5.433 | -0.037 0.035 0.001 -0.432 |
## healthy 0.216 8.043 | 0.179 1.853 0.075 4.730 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## sex | 0.095 0.070 0.449 |
## age_Q | 0.433 0.534 0.522 |
## spirituality | 0.199 0.099 0.001 |
## healthy | 0.142 0.216 0.075 |
## slimming | 0.100 0.272 0.107 |
## relaxing | 0.431 0.111 0.058 |# draw the MCA biplot
fviz_mca_biplot(mca_tea_reason, col.ind = "cos2", col.var = "red", alpha.var = "contrib", label = "var", geom = c("point","text"), labelsize = 3.5, arrows = c(FALSE,TRUE)) + labs(title = "MCA of tea consumption reasons") + theme_grey() + theme(axis.line = element_line(size = 0.5), panel.background = element_rect(fill = "gray93"))
The biplot shows interesting results. Variable categories which are very close show higher similarity. The plot includes two colour schemes representing the “cos2” of the individuals (as closer to 1 as better it is projected on the dimension) and the contribution of the variables to the dimenstions(“contrib”). The biplot gives following information:
- Men in the age of 25-34 drink tea not for health or weight loss reasons
- People in the age class of 45-59 show similarities with no relaxation and no spirituality reason for drinking tea
- People in the age class of 15-24, spirituality and relaxation reason are close and thereby similar
- Women & age class of 60+ have similarities with drinking tea for health and weight loosing reasons
Chapter 6: Analysis of longitudinal data
Data wrangling and performing an analysis of longitudinal data
Work of week 49 (02.12. - 08.12.2019)
The data wrangling of two data sets (bprs & rats) was performed before. The script of the data wrangling exercise can be found here.
1. Load and check the data sets
1.1. Load the wide (bprs & rats) and long (bprs_l & rats_l) data sets
# load necessary packages
library(tidyr)
library(dplyr)
library(corrplot)
library(ggplot2)
library(ggthemes)
library(GGally)
library(knitr)
library(kableExtra)
library(stringr)
library(lme4)# load the wide data sets
bprs <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep = " ", header = TRUE)
rats <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", header = TRUE)
# load the long data sets
bprs_l <- read.table("C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/bprs_l.txt")
rats_l <- read.table("C:/Users/richla/OneDrive/1 C - R-Folder/11-IODS-course/IODS-project/data/rats_l.txt")
# check the data sets
str(bprs_l) ## 'data.frame': 360 obs. of 5 variables:
## $ treatment: int 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : int 1 2 3 4 5 6 7 8 9 10 ...
## $ weeks : Factor w/ 9 levels "week0","week1",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ bprs : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week : int 0 0 0 0 0 0 0 0 0 0 ...str(rats_l)## 'data.frame': 176 obs. of 5 variables:
## $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : int 1 1 1 1 1 1 1 1 2 2 ...
## $ WD : Factor w/ 11 levels "WD1","WD15","WD22",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Weight: int 240 225 245 260 255 260 275 245 410 405 ...
## $ Time : int 1 1 1 1 1 1 1 1 1 1 ...The data sets show that the categorical variables “treatmeant” and “subject” in the BPRS data set and the categorical variables “Group” and “ID” in the RATS data set were read as numbers. So I factorized them again. See the code below.
# bprs data --> factor treatment & subject
bprs_l$treatment <- factor(bprs_l$treatment)
bprs_l$subject <- factor(bprs_l$subject)
# rats data --> factor ID & Group
rats_l$ID <- factor(rats_l$ID)
rats_l$Group <- factor(rats_l$Group)
# check the data again
str(bprs_l)## 'data.frame': 360 obs. of 5 variables:
## $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ weeks : Factor w/ 9 levels "week0","week1",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ bprs : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week : int 0 0 0 0 0 0 0 0 0 0 ...str(rats_l)## 'data.frame': 176 obs. of 5 variables:
## $ ID : Factor w/ 16 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 2 2 ...
## $ WD : Factor w/ 11 levels "WD1","WD15","WD22",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Weight: int 240 225 245 260 255 260 275 245 410 405 ...
## $ Time : int 1 1 1 1 1 1 1 1 1 1 ...1.2.The long data sets (bprs_l & rats_l)
The BPRS data set:
The BPRS (brief psychiatric rating scale) was measured on 40 male subjects with 2 treatment groups. One BPRS measurement was done before treatment started (week0), then weekly measurements followed up to 8 weeks in total. The BPRS assesses according to Vehkalahti & Everitt, 2019 the level of 18 symptom constructs which are used to evaluate if patients have schizophrenia. The BRRS symptoms constructs are measured in rates from 1 (not present) to 7 (extremely severe). The “long” BPRS data set bprs_l consists of 5 variables with 360 observations. These variables include:
- treatment - the psychological treatment the male subjects (treatment 1 & 2)
- subject - the male individuals identification number
- weeks - the factor variable showing the week of treatment and BPRS evaluation
- bprs - the bprs values
- week - the week number of the BPRS evaluation
# bprs_l data set
knitr::kable(bprs_l, caption = "BPRS - long data set") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>%
scroll_box(height = "300px")| treatment | subject | weeks | bprs | week |
|---|---|---|---|---|
| 1 | 1 | week0 | 42 | 0 |
| 1 | 2 | week0 | 58 | 0 |
| 1 | 3 | week0 | 54 | 0 |
| 1 | 4 | week0 | 55 | 0 |
| 1 | 5 | week0 | 72 | 0 |
| 1 | 6 | week0 | 48 | 0 |
| 1 | 7 | week0 | 71 | 0 |
| 1 | 8 | week0 | 30 | 0 |
| 1 | 9 | week0 | 41 | 0 |
| 1 | 10 | week0 | 57 | 0 |
| 1 | 11 | week0 | 30 | 0 |
| 1 | 12 | week0 | 55 | 0 |
| 1 | 13 | week0 | 36 | 0 |
| 1 | 14 | week0 | 38 | 0 |
| 1 | 15 | week0 | 66 | 0 |
| 1 | 16 | week0 | 41 | 0 |
| 1 | 17 | week0 | 45 | 0 |
| 1 | 18 | week0 | 39 | 0 |
| 1 | 19 | week0 | 24 | 0 |
| 1 | 20 | week0 | 38 | 0 |
| 2 | 1 | week0 | 52 | 0 |
| 2 | 2 | week0 | 30 | 0 |
| 2 | 3 | week0 | 65 | 0 |
| 2 | 4 | week0 | 37 | 0 |
| 2 | 5 | week0 | 59 | 0 |
| 2 | 6 | week0 | 30 | 0 |
| 2 | 7 | week0 | 69 | 0 |
| 2 | 8 | week0 | 62 | 0 |
| 2 | 9 | week0 | 38 | 0 |
| 2 | 10 | week0 | 65 | 0 |
| 2 | 11 | week0 | 78 | 0 |
| 2 | 12 | week0 | 38 | 0 |
| 2 | 13 | week0 | 63 | 0 |
| 2 | 14 | week0 | 40 | 0 |
| 2 | 15 | week0 | 40 | 0 |
| 2 | 16 | week0 | 54 | 0 |
| 2 | 17 | week0 | 33 | 0 |
| 2 | 18 | week0 | 28 | 0 |
| 2 | 19 | week0 | 52 | 0 |
| 2 | 20 | week0 | 47 | 0 |
| 1 | 1 | week1 | 36 | 1 |
| 1 | 2 | week1 | 68 | 1 |
| 1 | 3 | week1 | 55 | 1 |
| 1 | 4 | week1 | 77 | 1 |
| 1 | 5 | week1 | 75 | 1 |
| 1 | 6 | week1 | 43 | 1 |
| 1 | 7 | week1 | 61 | 1 |
| 1 | 8 | week1 | 36 | 1 |
| 1 | 9 | week1 | 43 | 1 |
| 1 | 10 | week1 | 51 | 1 |
| 1 | 11 | week1 | 34 | 1 |
| 1 | 12 | week1 | 52 | 1 |
| 1 | 13 | week1 | 32 | 1 |
| 1 | 14 | week1 | 35 | 1 |
| 1 | 15 | week1 | 68 | 1 |
| 1 | 16 | week1 | 35 | 1 |
| 1 | 17 | week1 | 38 | 1 |
| 1 | 18 | week1 | 35 | 1 |
| 1 | 19 | week1 | 28 | 1 |
| 1 | 20 | week1 | 34 | 1 |
| 2 | 1 | week1 | 73 | 1 |
| 2 | 2 | week1 | 23 | 1 |
| 2 | 3 | week1 | 31 | 1 |
| 2 | 4 | week1 | 31 | 1 |
| 2 | 5 | week1 | 67 | 1 |
| 2 | 6 | week1 | 33 | 1 |
| 2 | 7 | week1 | 52 | 1 |
| 2 | 8 | week1 | 54 | 1 |
| 2 | 9 | week1 | 40 | 1 |
| 2 | 10 | week1 | 44 | 1 |
| 2 | 11 | week1 | 95 | 1 |
| 2 | 12 | week1 | 41 | 1 |
| 2 | 13 | week1 | 65 | 1 |
| 2 | 14 | week1 | 37 | 1 |
| 2 | 15 | week1 | 36 | 1 |
| 2 | 16 | week1 | 45 | 1 |
| 2 | 17 | week1 | 41 | 1 |
| 2 | 18 | week1 | 30 | 1 |
| 2 | 19 | week1 | 43 | 1 |
| 2 | 20 | week1 | 36 | 1 |
| 1 | 1 | week2 | 36 | 2 |
| 1 | 2 | week2 | 61 | 2 |
| 1 | 3 | week2 | 41 | 2 |
| 1 | 4 | week2 | 49 | 2 |
| 1 | 5 | week2 | 72 | 2 |
| 1 | 6 | week2 | 41 | 2 |
| 1 | 7 | week2 | 47 | 2 |
| 1 | 8 | week2 | 38 | 2 |
| 1 | 9 | week2 | 39 | 2 |
| 1 | 10 | week2 | 51 | 2 |
| 1 | 11 | week2 | 34 | 2 |
| 1 | 12 | week2 | 49 | 2 |
| 1 | 13 | week2 | 36 | 2 |
| 1 | 14 | week2 | 36 | 2 |
| 1 | 15 | week2 | 65 | 2 |
| 1 | 16 | week2 | 45 | 2 |
| 1 | 17 | week2 | 46 | 2 |
| 1 | 18 | week2 | 27 | 2 |
| 1 | 19 | week2 | 31 | 2 |
| 1 | 20 | week2 | 27 | 2 |
| 2 | 1 | week2 | 42 | 2 |
| 2 | 2 | week2 | 32 | 2 |
| 2 | 3 | week2 | 33 | 2 |
| 2 | 4 | week2 | 27 | 2 |
| 2 | 5 | week2 | 58 | 2 |
| 2 | 6 | week2 | 37 | 2 |
| 2 | 7 | week2 | 41 | 2 |
| 2 | 8 | week2 | 49 | 2 |
| 2 | 9 | week2 | 38 | 2 |
| 2 | 10 | week2 | 31 | 2 |
| 2 | 11 | week2 | 75 | 2 |
| 2 | 12 | week2 | 36 | 2 |
| 2 | 13 | week2 | 60 | 2 |
| 2 | 14 | week2 | 31 | 2 |
| 2 | 15 | week2 | 55 | 2 |
| 2 | 16 | week2 | 35 | 2 |
| 2 | 17 | week2 | 30 | 2 |
| 2 | 18 | week2 | 29 | 2 |
| 2 | 19 | week2 | 26 | 2 |
| 2 | 20 | week2 | 32 | 2 |
| 1 | 1 | week3 | 43 | 3 |
| 1 | 2 | week3 | 55 | 3 |
| 1 | 3 | week3 | 38 | 3 |
| 1 | 4 | week3 | 54 | 3 |
| 1 | 5 | week3 | 65 | 3 |
| 1 | 6 | week3 | 38 | 3 |
| 1 | 7 | week3 | 30 | 3 |
| 1 | 8 | week3 | 38 | 3 |
| 1 | 9 | week3 | 35 | 3 |
| 1 | 10 | week3 | 55 | 3 |
| 1 | 11 | week3 | 41 | 3 |
| 1 | 12 | week3 | 54 | 3 |
| 1 | 13 | week3 | 31 | 3 |
| 1 | 14 | week3 | 34 | 3 |
| 1 | 15 | week3 | 49 | 3 |
| 1 | 16 | week3 | 42 | 3 |
| 1 | 17 | week3 | 38 | 3 |
| 1 | 18 | week3 | 25 | 3 |
| 1 | 19 | week3 | 28 | 3 |
| 1 | 20 | week3 | 25 | 3 |
| 2 | 1 | week3 | 41 | 3 |
| 2 | 2 | week3 | 24 | 3 |
| 2 | 3 | week3 | 28 | 3 |
| 2 | 4 | week3 | 31 | 3 |
| 2 | 5 | week3 | 61 | 3 |
| 2 | 6 | week3 | 33 | 3 |
| 2 | 7 | week3 | 33 | 3 |
| 2 | 8 | week3 | 39 | 3 |
| 2 | 9 | week3 | 27 | 3 |
| 2 | 10 | week3 | 34 | 3 |
| 2 | 11 | week3 | 76 | 3 |
| 2 | 12 | week3 | 27 | 3 |
| 2 | 13 | week3 | 53 | 3 |
| 2 | 14 | week3 | 38 | 3 |
| 2 | 15 | week3 | 55 | 3 |
| 2 | 16 | week3 | 27 | 3 |
| 2 | 17 | week3 | 32 | 3 |
| 2 | 18 | week3 | 33 | 3 |
| 2 | 19 | week3 | 27 | 3 |
| 2 | 20 | week3 | 29 | 3 |
| 1 | 1 | week4 | 41 | 4 |
| 1 | 2 | week4 | 43 | 4 |
| 1 | 3 | week4 | 43 | 4 |
| 1 | 4 | week4 | 56 | 4 |
| 1 | 5 | week4 | 50 | 4 |
| 1 | 6 | week4 | 36 | 4 |
| 1 | 7 | week4 | 27 | 4 |
| 1 | 8 | week4 | 31 | 4 |
| 1 | 9 | week4 | 28 | 4 |
| 1 | 10 | week4 | 53 | 4 |
| 1 | 11 | week4 | 36 | 4 |
| 1 | 12 | week4 | 48 | 4 |
| 1 | 13 | week4 | 25 | 4 |
| 1 | 14 | week4 | 25 | 4 |
| 1 | 15 | week4 | 36 | 4 |
| 1 | 16 | week4 | 31 | 4 |
| 1 | 17 | week4 | 40 | 4 |
| 1 | 18 | week4 | 29 | 4 |
| 1 | 19 | week4 | 29 | 4 |
| 1 | 20 | week4 | 25 | 4 |
| 2 | 1 | week4 | 39 | 4 |
| 2 | 2 | week4 | 20 | 4 |
| 2 | 3 | week4 | 22 | 4 |
| 2 | 4 | week4 | 31 | 4 |
| 2 | 5 | week4 | 49 | 4 |
| 2 | 6 | week4 | 28 | 4 |
| 2 | 7 | week4 | 34 | 4 |
| 2 | 8 | week4 | 55 | 4 |
| 2 | 9 | week4 | 31 | 4 |
| 2 | 10 | week4 | 39 | 4 |
| 2 | 11 | week4 | 66 | 4 |
| 2 | 12 | week4 | 29 | 4 |
| 2 | 13 | week4 | 52 | 4 |
| 2 | 14 | week4 | 35 | 4 |
| 2 | 15 | week4 | 42 | 4 |
| 2 | 16 | week4 | 25 | 4 |
| 2 | 17 | week4 | 46 | 4 |
| 2 | 18 | week4 | 30 | 4 |
| 2 | 19 | week4 | 24 | 4 |
| 2 | 20 | week4 | 25 | 4 |
| 1 | 1 | week5 | 40 | 5 |
| 1 | 2 | week5 | 34 | 5 |
| 1 | 3 | week5 | 28 | 5 |
| 1 | 4 | week5 | 50 | 5 |
| 1 | 5 | week5 | 39 | 5 |
| 1 | 6 | week5 | 29 | 5 |
| 1 | 7 | week5 | 40 | 5 |
| 1 | 8 | week5 | 26 | 5 |
| 1 | 9 | week5 | 22 | 5 |
| 1 | 10 | week5 | 43 | 5 |
| 1 | 11 | week5 | 36 | 5 |
| 1 | 12 | week5 | 43 | 5 |
| 1 | 13 | week5 | 25 | 5 |
| 1 | 14 | week5 | 27 | 5 |
| 1 | 15 | week5 | 32 | 5 |
| 1 | 16 | week5 | 31 | 5 |
| 1 | 17 | week5 | 33 | 5 |
| 1 | 18 | week5 | 28 | 5 |
| 1 | 19 | week5 | 21 | 5 |
| 1 | 20 | week5 | 27 | 5 |
| 2 | 1 | week5 | 38 | 5 |
| 2 | 2 | week5 | 20 | 5 |
| 2 | 3 | week5 | 25 | 5 |
| 2 | 4 | week5 | 26 | 5 |
| 2 | 5 | week5 | 38 | 5 |
| 2 | 6 | week5 | 26 | 5 |
| 2 | 7 | week5 | 37 | 5 |
| 2 | 8 | week5 | 51 | 5 |
| 2 | 9 | week5 | 24 | 5 |
| 2 | 10 | week5 | 34 | 5 |
| 2 | 11 | week5 | 64 | 5 |
| 2 | 12 | week5 | 27 | 5 |
| 2 | 13 | week5 | 32 | 5 |
| 2 | 14 | week5 | 30 | 5 |
| 2 | 15 | week5 | 30 | 5 |
| 2 | 16 | week5 | 22 | 5 |
| 2 | 17 | week5 | 43 | 5 |
| 2 | 18 | week5 | 26 | 5 |
| 2 | 19 | week5 | 32 | 5 |
| 2 | 20 | week5 | 23 | 5 |
| 1 | 1 | week6 | 38 | 6 |
| 1 | 2 | week6 | 28 | 6 |
| 1 | 3 | week6 | 29 | 6 |
| 1 | 4 | week6 | 47 | 6 |
| 1 | 5 | week6 | 32 | 6 |
| 1 | 6 | week6 | 33 | 6 |
| 1 | 7 | week6 | 30 | 6 |
| 1 | 8 | week6 | 26 | 6 |
| 1 | 9 | week6 | 20 | 6 |
| 1 | 10 | week6 | 43 | 6 |
| 1 | 11 | week6 | 38 | 6 |
| 1 | 12 | week6 | 37 | 6 |
| 1 | 13 | week6 | 21 | 6 |
| 1 | 14 | week6 | 25 | 6 |
| 1 | 15 | week6 | 27 | 6 |
| 1 | 16 | week6 | 29 | 6 |
| 1 | 17 | week6 | 27 | 6 |
| 1 | 18 | week6 | 21 | 6 |
| 1 | 19 | week6 | 22 | 6 |
| 1 | 20 | week6 | 21 | 6 |
| 2 | 1 | week6 | 43 | 6 |
| 2 | 2 | week6 | 19 | 6 |
| 2 | 3 | week6 | 24 | 6 |
| 2 | 4 | week6 | 24 | 6 |
| 2 | 5 | week6 | 37 | 6 |
| 2 | 6 | week6 | 27 | 6 |
| 2 | 7 | week6 | 37 | 6 |
| 2 | 8 | week6 | 55 | 6 |
| 2 | 9 | week6 | 22 | 6 |
| 2 | 10 | week6 | 41 | 6 |
| 2 | 11 | week6 | 64 | 6 |
| 2 | 12 | week6 | 21 | 6 |
| 2 | 13 | week6 | 37 | 6 |
| 2 | 14 | week6 | 33 | 6 |
| 2 | 15 | week6 | 26 | 6 |
| 2 | 16 | week6 | 22 | 6 |
| 2 | 17 | week6 | 43 | 6 |
| 2 | 18 | week6 | 36 | 6 |
| 2 | 19 | week6 | 21 | 6 |
| 2 | 20 | week6 | 23 | 6 |
| 1 | 1 | week7 | 47 | 7 |
| 1 | 2 | week7 | 28 | 7 |
| 1 | 3 | week7 | 25 | 7 |
| 1 | 4 | week7 | 42 | 7 |
| 1 | 5 | week7 | 38 | 7 |
| 1 | 6 | week7 | 27 | 7 |
| 1 | 7 | week7 | 31 | 7 |
| 1 | 8 | week7 | 25 | 7 |
| 1 | 9 | week7 | 23 | 7 |
| 1 | 10 | week7 | 39 | 7 |
| 1 | 11 | week7 | 36 | 7 |
| 1 | 12 | week7 | 36 | 7 |
| 1 | 13 | week7 | 19 | 7 |
| 1 | 14 | week7 | 26 | 7 |
| 1 | 15 | week7 | 30 | 7 |
| 1 | 16 | week7 | 26 | 7 |
| 1 | 17 | week7 | 31 | 7 |
| 1 | 18 | week7 | 25 | 7 |
| 1 | 19 | week7 | 23 | 7 |
| 1 | 20 | week7 | 19 | 7 |
| 2 | 1 | week7 | 62 | 7 |
| 2 | 2 | week7 | 18 | 7 |
| 2 | 3 | week7 | 31 | 7 |
| 2 | 4 | week7 | 26 | 7 |
| 2 | 5 | week7 | 36 | 7 |
| 2 | 6 | week7 | 23 | 7 |
| 2 | 7 | week7 | 38 | 7 |
| 2 | 8 | week7 | 59 | 7 |
| 2 | 9 | week7 | 21 | 7 |
| 2 | 10 | week7 | 42 | 7 |
| 2 | 11 | week7 | 60 | 7 |
| 2 | 12 | week7 | 22 | 7 |
| 2 | 13 | week7 | 52 | 7 |
| 2 | 14 | week7 | 30 | 7 |
| 2 | 15 | week7 | 30 | 7 |
| 2 | 16 | week7 | 22 | 7 |
| 2 | 17 | week7 | 43 | 7 |
| 2 | 18 | week7 | 33 | 7 |
| 2 | 19 | week7 | 21 | 7 |
| 2 | 20 | week7 | 23 | 7 |
| 1 | 1 | week8 | 51 | 8 |
| 1 | 2 | week8 | 28 | 8 |
| 1 | 3 | week8 | 24 | 8 |
| 1 | 4 | week8 | 46 | 8 |
| 1 | 5 | week8 | 32 | 8 |
| 1 | 6 | week8 | 25 | 8 |
| 1 | 7 | week8 | 31 | 8 |
| 1 | 8 | week8 | 24 | 8 |
| 1 | 9 | week8 | 21 | 8 |
| 1 | 10 | week8 | 32 | 8 |
| 1 | 11 | week8 | 36 | 8 |
| 1 | 12 | week8 | 31 | 8 |
| 1 | 13 | week8 | 22 | 8 |
| 1 | 14 | week8 | 26 | 8 |
| 1 | 15 | week8 | 37 | 8 |
| 1 | 16 | week8 | 30 | 8 |
| 1 | 17 | week8 | 27 | 8 |
| 1 | 18 | week8 | 20 | 8 |
| 1 | 19 | week8 | 22 | 8 |
| 1 | 20 | week8 | 21 | 8 |
| 2 | 1 | week8 | 50 | 8 |
| 2 | 2 | week8 | 20 | 8 |
| 2 | 3 | week8 | 32 | 8 |
| 2 | 4 | week8 | 23 | 8 |
| 2 | 5 | week8 | 35 | 8 |
| 2 | 6 | week8 | 21 | 8 |
| 2 | 7 | week8 | 35 | 8 |
| 2 | 8 | week8 | 66 | 8 |
| 2 | 9 | week8 | 21 | 8 |
| 2 | 10 | week8 | 39 | 8 |
| 2 | 11 | week8 | 75 | 8 |
| 2 | 12 | week8 | 23 | 8 |
| 2 | 13 | week8 | 28 | 8 |
| 2 | 14 | week8 | 27 | 8 |
| 2 | 15 | week8 | 37 | 8 |
| 2 | 16 | week8 | 22 | 8 |
| 2 | 17 | week8 | 43 | 8 |
| 2 | 18 | week8 | 30 | 8 |
| 2 | 19 | week8 | 21 | 8 |
| 2 | 20 | week8 | 23 | 8 |
The RATS data set:
This data consists of measurements on how rats grow. There were 3 groups of rats defined which got a different diet and the body weight of each animal was measured over a period of 9 weeks. The research question is if the growth profiles of the different diet groups differ (Vehkalahti & Everitt ,2019). The “long” RATS data set rats_l consists of 5 variables with 176 observations. The variables include:
- ID - the individual rat’s identification number
- Group - the diet group (1, 2 & 3)
- WD - the factorial value of the time (the day when the rats’ weights were measured)
- Weight - weight of each individual rat at the given day
- Time - the time point of the weight measurement (days)
# rats_ data set
knitr::kable(rats_l, caption = "RATS - long data set") %>%
kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>%
scroll_box(height = "300px")| ID | Group | WD | Weight | Time |
|---|---|---|---|---|
| 1 | 1 | WD1 | 240 | 1 |
| 2 | 1 | WD1 | 225 | 1 |
| 3 | 1 | WD1 | 245 | 1 |
| 4 | 1 | WD1 | 260 | 1 |
| 5 | 1 | WD1 | 255 | 1 |
| 6 | 1 | WD1 | 260 | 1 |
| 7 | 1 | WD1 | 275 | 1 |
| 8 | 1 | WD1 | 245 | 1 |
| 9 | 2 | WD1 | 410 | 1 |
| 10 | 2 | WD1 | 405 | 1 |
| 11 | 2 | WD1 | 445 | 1 |
| 12 | 2 | WD1 | 555 | 1 |
| 13 | 3 | WD1 | 470 | 1 |
| 14 | 3 | WD1 | 535 | 1 |
| 15 | 3 | WD1 | 520 | 1 |
| 16 | 3 | WD1 | 510 | 1 |
| 1 | 1 | WD8 | 250 | 8 |
| 2 | 1 | WD8 | 230 | 8 |
| 3 | 1 | WD8 | 250 | 8 |
| 4 | 1 | WD8 | 255 | 8 |
| 5 | 1 | WD8 | 260 | 8 |
| 6 | 1 | WD8 | 265 | 8 |
| 7 | 1 | WD8 | 275 | 8 |
| 8 | 1 | WD8 | 255 | 8 |
| 9 | 2 | WD8 | 415 | 8 |
| 10 | 2 | WD8 | 420 | 8 |
| 11 | 2 | WD8 | 445 | 8 |
| 12 | 2 | WD8 | 560 | 8 |
| 13 | 3 | WD8 | 465 | 8 |
| 14 | 3 | WD8 | 525 | 8 |
| 15 | 3 | WD8 | 525 | 8 |
| 16 | 3 | WD8 | 510 | 8 |
| 1 | 1 | WD15 | 255 | 15 |
| 2 | 1 | WD15 | 230 | 15 |
| 3 | 1 | WD15 | 250 | 15 |
| 4 | 1 | WD15 | 255 | 15 |
| 5 | 1 | WD15 | 255 | 15 |
| 6 | 1 | WD15 | 270 | 15 |
| 7 | 1 | WD15 | 260 | 15 |
| 8 | 1 | WD15 | 260 | 15 |
| 9 | 2 | WD15 | 425 | 15 |
| 10 | 2 | WD15 | 430 | 15 |
| 11 | 2 | WD15 | 450 | 15 |
| 12 | 2 | WD15 | 565 | 15 |
| 13 | 3 | WD15 | 475 | 15 |
| 14 | 3 | WD15 | 530 | 15 |
| 15 | 3 | WD15 | 530 | 15 |
| 16 | 3 | WD15 | 520 | 15 |
| 1 | 1 | WD22 | 260 | 22 |
| 2 | 1 | WD22 | 232 | 22 |
| 3 | 1 | WD22 | 255 | 22 |
| 4 | 1 | WD22 | 265 | 22 |
| 5 | 1 | WD22 | 270 | 22 |
| 6 | 1 | WD22 | 275 | 22 |
| 7 | 1 | WD22 | 270 | 22 |
| 8 | 1 | WD22 | 268 | 22 |
| 9 | 2 | WD22 | 428 | 22 |
| 10 | 2 | WD22 | 440 | 22 |
| 11 | 2 | WD22 | 452 | 22 |
| 12 | 2 | WD22 | 580 | 22 |
| 13 | 3 | WD22 | 485 | 22 |
| 14 | 3 | WD22 | 533 | 22 |
| 15 | 3 | WD22 | 540 | 22 |
| 16 | 3 | WD22 | 515 | 22 |
| 1 | 1 | WD29 | 262 | 29 |
| 2 | 1 | WD29 | 240 | 29 |
| 3 | 1 | WD29 | 262 | 29 |
| 4 | 1 | WD29 | 265 | 29 |
| 5 | 1 | WD29 | 270 | 29 |
| 6 | 1 | WD29 | 275 | 29 |
| 7 | 1 | WD29 | 273 | 29 |
| 8 | 1 | WD29 | 270 | 29 |
| 9 | 2 | WD29 | 438 | 29 |
| 10 | 2 | WD29 | 448 | 29 |
| 11 | 2 | WD29 | 455 | 29 |
| 12 | 2 | WD29 | 590 | 29 |
| 13 | 3 | WD29 | 487 | 29 |
| 14 | 3 | WD29 | 535 | 29 |
| 15 | 3 | WD29 | 543 | 29 |
| 16 | 3 | WD29 | 530 | 29 |
| 1 | 1 | WD36 | 258 | 36 |
| 2 | 1 | WD36 | 240 | 36 |
| 3 | 1 | WD36 | 265 | 36 |
| 4 | 1 | WD36 | 268 | 36 |
| 5 | 1 | WD36 | 273 | 36 |
| 6 | 1 | WD36 | 277 | 36 |
| 7 | 1 | WD36 | 274 | 36 |
| 8 | 1 | WD36 | 265 | 36 |
| 9 | 2 | WD36 | 443 | 36 |
| 10 | 2 | WD36 | 460 | 36 |
| 11 | 2 | WD36 | 455 | 36 |
| 12 | 2 | WD36 | 597 | 36 |
| 13 | 3 | WD36 | 493 | 36 |
| 14 | 3 | WD36 | 540 | 36 |
| 15 | 3 | WD36 | 546 | 36 |
| 16 | 3 | WD36 | 538 | 36 |
| 1 | 1 | WD43 | 266 | 43 |
| 2 | 1 | WD43 | 243 | 43 |
| 3 | 1 | WD43 | 267 | 43 |
| 4 | 1 | WD43 | 270 | 43 |
| 5 | 1 | WD43 | 274 | 43 |
| 6 | 1 | WD43 | 278 | 43 |
| 7 | 1 | WD43 | 276 | 43 |
| 8 | 1 | WD43 | 265 | 43 |
| 9 | 2 | WD43 | 442 | 43 |
| 10 | 2 | WD43 | 458 | 43 |
| 11 | 2 | WD43 | 451 | 43 |
| 12 | 2 | WD43 | 595 | 43 |
| 13 | 3 | WD43 | 493 | 43 |
| 14 | 3 | WD43 | 525 | 43 |
| 15 | 3 | WD43 | 538 | 43 |
| 16 | 3 | WD43 | 535 | 43 |
| 1 | 1 | WD44 | 266 | 44 |
| 2 | 1 | WD44 | 244 | 44 |
| 3 | 1 | WD44 | 267 | 44 |
| 4 | 1 | WD44 | 272 | 44 |
| 5 | 1 | WD44 | 273 | 44 |
| 6 | 1 | WD44 | 278 | 44 |
| 7 | 1 | WD44 | 271 | 44 |
| 8 | 1 | WD44 | 267 | 44 |
| 9 | 2 | WD44 | 446 | 44 |
| 10 | 2 | WD44 | 464 | 44 |
| 11 | 2 | WD44 | 450 | 44 |
| 12 | 2 | WD44 | 595 | 44 |
| 13 | 3 | WD44 | 504 | 44 |
| 14 | 3 | WD44 | 530 | 44 |
| 15 | 3 | WD44 | 544 | 44 |
| 16 | 3 | WD44 | 542 | 44 |
| 1 | 1 | WD50 | 265 | 50 |
| 2 | 1 | WD50 | 238 | 50 |
| 3 | 1 | WD50 | 264 | 50 |
| 4 | 1 | WD50 | 274 | 50 |
| 5 | 1 | WD50 | 276 | 50 |
| 6 | 1 | WD50 | 284 | 50 |
| 7 | 1 | WD50 | 282 | 50 |
| 8 | 1 | WD50 | 273 | 50 |
| 9 | 2 | WD50 | 456 | 50 |
| 10 | 2 | WD50 | 475 | 50 |
| 11 | 2 | WD50 | 462 | 50 |
| 12 | 2 | WD50 | 612 | 50 |
| 13 | 3 | WD50 | 507 | 50 |
| 14 | 3 | WD50 | 543 | 50 |
| 15 | 3 | WD50 | 553 | 50 |
| 16 | 3 | WD50 | 550 | 50 |
| 1 | 1 | WD57 | 272 | 57 |
| 2 | 1 | WD57 | 247 | 57 |
| 3 | 1 | WD57 | 268 | 57 |
| 4 | 1 | WD57 | 273 | 57 |
| 5 | 1 | WD57 | 278 | 57 |
| 6 | 1 | WD57 | 279 | 57 |
| 7 | 1 | WD57 | 281 | 57 |
| 8 | 1 | WD57 | 274 | 57 |
| 9 | 2 | WD57 | 468 | 57 |
| 10 | 2 | WD57 | 484 | 57 |
| 11 | 2 | WD57 | 466 | 57 |
| 12 | 2 | WD57 | 618 | 57 |
| 13 | 3 | WD57 | 518 | 57 |
| 14 | 3 | WD57 | 544 | 57 |
| 15 | 3 | WD57 | 555 | 57 |
| 16 | 3 | WD57 | 553 | 57 |
| 1 | 1 | WD64 | 278 | 64 |
| 2 | 1 | WD64 | 245 | 64 |
| 3 | 1 | WD64 | 269 | 64 |
| 4 | 1 | WD64 | 275 | 64 |
| 5 | 1 | WD64 | 280 | 64 |
| 6 | 1 | WD64 | 281 | 64 |
| 7 | 1 | WD64 | 284 | 64 |
| 8 | 1 | WD64 | 278 | 64 |
| 9 | 2 | WD64 | 478 | 64 |
| 10 | 2 | WD64 | 496 | 64 |
| 11 | 2 | WD64 | 472 | 64 |
| 12 | 2 | WD64 | 628 | 64 |
| 13 | 3 | WD64 | 525 | 64 |
| 14 | 3 | WD64 | 559 | 64 |
| 15 | 3 | WD64 | 548 | 64 |
| 16 | 3 | WD64 | 569 | 64 |
2. Analysis of the long data sets (bprs_l & rats_l)
The data set analyses presented in the MABS book and the DataCamp platform are swapped. So we run the data analyses of MABS book chapter 8 on the “rats_l” data set and the analyses of MABS book chapter 9 on the “bprs_l” data set.
3. Analyses of chapter 8 of MABS using the RATS data (rats_l)
The long rats_l data will be analysed according to the instructions from chapter 8 of MABS. We start as a first step with a graphical overview of the data.
3.1. A first plot
# plot 1
ggplot(rats_l, aes(x = Time, y = Weight,linetype = ID)) +
geom_line(color = "darkgreen") +
scale_linetype_manual(values = rep(1:6, times= 3)) + # there are 16 IDs (6 linetypes are available - so repeated 3 times)
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "right") +
scale_y_continuous(limits = c(min(rats_l$Weight), max(rats_l$Weight))) +
ggtitle("RATS: long data set ")
Here the rats_l data is plotted. The x-axis with the “Time” represents the days of the weight measurements, the y-axis shows the weights of the individual rats at the different measurement times. The plot is splitted into the three different diet groups. What can be seen in the graph is that the rat’s starting weights in group 2 & 3 were higher than in group 1 and that the weights in group 2 & 3 increased in quite high rates, while in group 1 it seems that the weights increased in a much smaller rate.
The data also shows the “tracking” phenomena (acc. MABS book, Vehkalahti & Everitt, 2019) where rats with a higher starting weight tend to have a higher weight increase over time. To make this tracking seen more clearly we standardise the rats_l data set.
3.2. Standardized data
The rats_l data set is standardised by calculating the “StdWeight”with following formular: \(StdWeight = (Weight - Mean(Weight)) / SD (Weight)\)
The calculated StdWeight is added as a new variable to the rats_l data set. Then the StdWeight is plotted.
# Standardise the variable Weight
rats_l <- rats_l %>%
group_by(Time) %>%
mutate(StdWeight = ((Weight - mean(Weight))/sd(Weight))) %>%
ungroup()
# Glimpse the data
glimpse(rats_l)## Observations: 176
## Variables: 6
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1...
## $ WD <fct> WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 44...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8...
## $ StdWeight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8...# Plot again with the standardised rats_l
ggplot(rats_l, aes(x = Time, y = StdWeight, linetype = ID)) +
geom_line(color = "darkred") +
scale_linetype_manual(values = rep(1:6, times=3)) +
facet_grid(. ~ Group, labeller = label_both) +
scale_y_continuous(name = "StdWeight - standardized Weight values") +
ggtitle("RATS: standardized Weight values") +
theme(panel.grid.major.y = element_line(colour = "grey40"))
The “StdWeight”-plot shows the standardized weight values over the measurement period. Here the “tracking” (higher starting weights tent to increase more compared to lower weights and stay on a higher value for the measurement period) can be seen much better.
3.3. Summary graphs
First you see an overview of the rats weights over the whole measurement period with boxplots. The different diet groups are seen in different colors.
# add the Time as a categorical variable "Time1"
rats_l <- rats_l %>%
mutate(Time1 = factor(rats_l$Time)) %>%
ungroup()
# check the data
str(rats_l)## Classes 'tbl_df', 'tbl' and 'data.frame': 176 obs. of 7 variables:
## $ ID : Factor w/ 16 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 2 2 ...
## $ WD : Factor w/ 11 levels "WD1","WD15","WD22",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Weight : int 240 225 245 260 255 260 275 245 410 405 ...
## $ Time : int 1 1 1 1 1 1 1 1 1 1 ...
## $ StdWeight: num -1.001 -1.12 -0.961 -0.842 -0.882 ...
## $ Time1 : Factor w/ 11 levels "1","8","15","22",..: 1 1 1 1 1 1 1 1 1 1 ...# prepare a boxplot of the "rats_l" data set
ggplot(data = rats_l, aes(x = rats_l$Time1, y = Weight, fill = rats_l$Group)) +
geom_boxplot() +
ylab("Weight") +
xlab("Weight measurement time points [days]") +
ggtitle("Rat weights over the measurment period") +
scale_fill_discrete(name = "Diet group") +
theme(legend.position = "right") 
Now, the sample number n_rats of the individually measured rat weights are calculated (in total 11 weight measurements, so n = 11). Then a new table rats_s is prepared, where the calculated mean weight of all rats & the standard error of each diet group is added as a variable. The standard error is calculated via following formular: \(SE = SD(Weight)/ \sqrt n\) This new data table is then presented in the next plot.
# Number of Time, baseline (Time = 1) included
n_rats <- rats_l$Time %>% unique() %>% length()
# Summary data with mean and standard error of rats_l by group and time
rats_s <- rats_l %>%
group_by(Group, Time) %>%
summarise(mean = mean(Weight), se = (sd(Weight)/sqrt(n_rats))) %>%
ungroup()
# Glimpse the data
str(rats_s)## Classes 'tbl_df', 'tbl' and 'data.frame': 33 obs. of 4 variables:
## $ Group: Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
## $ Time : int 1 8 15 22 29 36 43 44 50 57 ...
## $ mean : num 251 255 254 262 265 ...
## $ se : num 4.59 3.95 3.46 4.1 3.33 ...# Plot the mean profiles
ggplot(rats_s, aes(x = Time, y = mean, linetype = Group, color = Group, shape = Group)) +
geom_line(size = 0.6) +
scale_linetype_manual(name = "Diet group", values = c(1,2,3)) +
geom_point(size=1.5) +
scale_shape_manual(name = "Diet group", values = c(16,17,18)) +
scale_color_manual(name = "Diet group", values = c("red", "darkgreen", "blue")) +
geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.8) +
theme(legend.position = "right") +
scale_y_continuous(name = "Mean (Weight) +/- SE (Weight)") +
ggtitle("Mean weight profiles of the different diet groups")
This graph shows the rat weight development of the three different diet groups over the measurement period. Group 1 starts on a lower level and the weight increases just slightly (~250 to 280). Group 2 and group 3 weights start form a higher values and also the rats’s weights decrease on a higher level (in group 2 an increase of more that 50 (+/- of course…), in group 3 around 50 (+/- a smaller value)). What you can see is that the standard errors in group 2 & 3 are much higher, so there is a much bigger weight difference between the individual rats within the diet group.
3.4. Find the outlier
Here we will have now a look how the rat’s weight developed after the first measurement. Therefore we prepare a new data table rats_l10s where the weight data without time = 1 is summarised by calculating the mean value of weight per diet group.
# Create a summary data by Group and ID with mean as the summary variable (ignoring baseline Time 1).
rats_l10s <- rats_l %>%
filter(Time > 1) %>%
group_by(Group, ID) %>%
summarise(mean=mean(Weight)) %>%
ungroup()
# Glimpse the data
glimpse(rats_l10s)## Observations: 16
## Variables: 3
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ mean <dbl> 263.2, 238.9, 261.7, 267.2, 270.9, 276.2, 274.6, 267.5, ...From this new data table we prepare a boxplot:
# Draw a boxplot of the mean versus group
ggplot(rats_l10s, aes(x = Group, y = mean, fill = Group)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape = 23, size = 3, fill = "black") +
scale_y_continuous(name = "Mean (Weight) / Time 8-64 [days]") +
xlab("Diet group") +
scale_fill_discrete(name = "Diet group") +
ggtitle("Mean weights of diet groups excluding the first measurement")
The boxplot shows the mean rat weights of each diet group of the measurements after the first measurement (day 1). The boxplots of all three diet groups actually show an outlier. These outliers can disturb conclusions from further comparisons of the diet groups. So we have to remove these outliers as I did it for each group seen in the code below. I filter out the outliers and create a new data table rats_l10s1 where the mean values without the outliers are collected. Then a new boxplot of the diet groups is prepared.
# Create a new data by filtering the outliers and adjust the ggplot code the draw the plot again with the new data
rats_l10s1 <- rats_l10s %>%
filter(
(mean > 250 & Group == 1) |
(mean < 550 & Group == 2) |
(mean > 500 & Group == 3)) %>%
ungroup()
str(rats_l10s1)## Classes 'tbl_df', 'tbl' and 'data.frame': 13 obs. of 3 variables:
## $ Group: Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 2 2 2 ...
## $ ID : Factor w/ 16 levels "1","2","3","4",..: 1 3 4 5 6 7 8 9 10 11 ...
## $ mean : num 263 262 267 271 276 ...# Draw a boxplot of the mean versus group with the outlier filtered data
ggplot(rats_l10s1, aes(x = Group, y = mean, fill = Group)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape = 23, size = 3, fill = "black") +
scale_y_continuous(name = "Mean (Weight) / Time 8-64 [days]") +
xlab("Diet group") +
scale_fill_discrete(name = "Diet group") +
ggtitle("Mean weights of diet groups excluding the first measurement")
Here we see the boxplot graph with the weights means over the measurement period 8 to 64 days without outlier values. With that data we continue to analyse the diet group differences.
3.5. T-test and ANOVA
We cannot perform a t-test between the groups since we have 3 groups and not 2 groups.
t.test(mean ~ rats_l10s2$Group == 2, paired = TRUE)
# Two-sample t-test group 1 & 3
t.test(mean ~ Group == 3, data = rats_l10s1)
# cannot be done on three groupsNow we continue with creating a new data table rats_l10s2where a so called “baseline” weight values are created from the first weigh measurement on day 1 (WD1). So each group weight baseline was defined. See the table below with the mean weight values of the whole measurement period and the starting weight
# Add the baseline from the original data as a new variable to the summary data
rats_l10s2 <- rats_l10s %>%
mutate(baseline = rats$WD1)
# check the data table
rats_l10s2 %>% knitr::kable() %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>% scroll_box(height = "300px")| Group | ID | mean | baseline |
|---|---|---|---|
| 1 | 1 | 263.2 | 240 |
| 1 | 2 | 238.9 | 225 |
| 1 | 3 | 261.7 | 245 |
| 1 | 4 | 267.2 | 260 |
| 1 | 5 | 270.9 | 255 |
| 1 | 6 | 276.2 | 260 |
| 1 | 7 | 274.6 | 275 |
| 1 | 8 | 267.5 | 245 |
| 2 | 9 | 443.9 | 410 |
| 2 | 10 | 457.5 | 405 |
| 2 | 11 | 455.8 | 445 |
| 2 | 12 | 594.0 | 555 |
| 3 | 13 | 495.2 | 470 |
| 3 | 14 | 536.4 | 535 |
| 3 | 15 | 542.2 | 520 |
| 3 | 16 | 536.2 | 510 |
Using this new data table with the mean weight values and the baseline weight values we perform an anova test to see which diet group had the highest weight increase over the measurement period. there are differences between the diet groups. In the following codes I run first a linear model comparing the group means to each other (from the data table rats_l10s2 excluding the starting weight measurement) and in the second linear model I compare the baseline weights of each group with the mean weight value of each group (also from the rats_l10s2 data table).
I continue now to compare the diet group differences with performing an ANOVA and a post-hoc test (TukeyHSD) to compare the group means, since I want to know if the diet groups are different from each other. The ANOVA is performed from the data table rats_l10s2. This is the data table without the first measurement (start value) and inluding the outlier values.
# linear regression model of mean values versus diet groups
rats_fit1 <- lm(mean ~ Group, data = rats_l10s2)
# regression model output
round(rats_fit1$coefficients, digits = 2) %>% knitr::kable() %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| x | |
|---|---|
| (Intercept) | 265.02 |
| Group2 | 222.77 |
| Group3 | 262.48 |
The model shows that the mean value of group 2 is 222g higher and the mean value of group 3 is 262g higher compared to group 1 mean weights. Let’s have a look now if the diet groups differ to each other. With a post-hoc test after the ANOVA we can check the group means.
# Compute the analysis of variance table of rats_fit1 <- lm(mean ~ Group, data = rats_l10s2) with anova()
rats_anova1 <- anova(rats_fit1)
rats_anova1## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## Group 2 238620 119310 88.525 2.679e-08 ***
## Residuals 13 17521 1348
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The ANOVA of the linear model of mean weight ~ diet group shows that group 2 has a significant different weight increase compared to group 1 & group 3.
# another kind of anova calculation --> compute the anova with aov()
rats_aov2 <- aov(mean ~ Group, data = rats_l10s2)
summary(rats_aov2)## Df Sum Sq Mean Sq F value Pr(>F)
## Group 2 238620 119310 88.53 2.68e-08 ***
## Residuals 13 17521 1348
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# compare the anova results with tukeyHSD
TukeyHSD(rats_aov2, "Group", ordered = TRUE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
## factor levels have been ordered
##
## Fit: aov(formula = mean ~ Group, data = rats_l10s2)
##
## $Group
## diff lwr upr p adj
## 2-1 222.775 163.41452 282.1355 0.0000006
## 3-1 262.475 203.11452 321.8355 0.0000001
## 3-2 39.700 -28.84358 108.2436 0.3099506Here we can see that the secon ANOVA function gives the same result and this result can be put in the TukeyHSD function. This post-hoc test gives us then the differences between the diet groups. Group 1 & 2 are significantly different in the mean weight. Group 1 & 3 are as well. Group 2 and 3 are not significantly different in the weight means (there is the outlier influence).
Now lets see what is the difference of each group to the baselines.
# Fit the linear model with the mean as the response
rats_fit2 <- lm(mean ~ baseline + Group, data = rats_l10s2)
# regression model output
round(rats_fit2$coefficients, digits = 2) %>% knitr::kable() %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center")| x | |
|---|---|
| (Intercept) | 33.16 |
| baseline | 0.93 |
| Group2 | 34.86 |
| Group3 | 23.68 |
Here a regression model of the mean rats weights compared to the weight baseline (the first weight measurement of each dietary group). The results shows that the weight difference between baseline and group 2 is around 34g and group 3 and baseline is around 23g different. So group 2 has actually the highest weight increase rates over the measurement period.
# Compute the analysis of variance table of rats_fit2 <- lm(mean ~ baseline + Group, data = rats_l10s2) with anova()
rats_anova2 <- anova(rats_fit2)
rats_anova2## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 253625 253625 1859.8201 1.57e-14 ***
## Group 2 879 439 3.2219 0.07586 .
## Residuals 12 1636 136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1What is seen from the second ANOVA is that the baseline has a significant influence on the weight increase later in the different diet groups and that group 2 has a rather good significant weight increase relative to the baseline compared to group 1 and group 3. So it seems that diet 2 leads to the highest weight increase.
4. Analyses of Chapter 9 of MABS using the BPRS data (bprs_l)
4.1. First plots of the data
Here follows now the analysis of BPRS data set with linear mixed models. The BPRS data includes measurements over a period of 8 weeks with a starting value 0 (so all in all 9 BPRS determinations).
str(bprs_l)## 'data.frame': 360 obs. of 5 variables:
## $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ weeks : Factor w/ 9 levels "week0","week1",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ bprs : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week : int 0 0 0 0 0 0 0 0 0 0 ...# overview plot
pairs(bprs_l, col = bprs_l$subject)
#plotting the bprs_l data 1
ggplot(bprs_l, aes(x = week, y = bprs, shape = subject, group = treatment)) +
geom_point(color = "darkgreen") +
scale_shape_manual(values = rep(1:10, times = 2)) +
scale_x_continuous(name = "Weeks", breaks = seq(0,8,1)) +
scale_y_continuous(name = "BPRS value") +
theme(legend.position = "bottom") +
ggtitle("BPRS: data values overview") +
theme(legend.box.background = element_rect(),legend.box.margin = margin(2, 2, 2, 2))
#plotting the bprs_l data 2
ggplot(bprs_l, aes(x = week, y = bprs, linetype = subject)) +
geom_line(color = "darkgreen") +
scale_linetype_manual(values = rep(1:6, times = 4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_x_continuous(name = "Weeks", breaks = seq(0,8,1)) +
scale_y_continuous(name = "BPRS values observed", breaks = seq(10,100,5)) +
theme(legend.position = "bottom") +
ggtitle("BPRS: data values overview by treatment") +
theme(legend.box.background = element_rect(),legend.box.margin = margin(2, 2, 2, 2))
#plotting the bprs_l data 3
ggplot(bprs_l, aes(x = week, y = bprs)) +
geom_line(aes(linetype = treatment), color = "darkgreen") +
scale_linetype_manual(values = rep(1:2, times = 1)) +
facet_grid(.~ subject) +
scale_x_continuous(name = "Weeks", breaks = seq(0,8,4)) +
scale_y_continuous(name = "BPRS values observed", breaks = seq(10,100,5)) +
theme(legend.position = "right") +
ggtitle("BPRS: data values overview by subject") +
theme(legend.box.background = element_rect(),legend.box.margin = margin(2, 2, 2, 2))
The overview plots give us a glimpse into the data. We have 40 subjects in two treatment groups (so 20 in each group). After a first BPRS determination (week 0) the treatment started and 8 more BPRS determinations followed.
4.2. Linear model of bprs data
We start with a first linear regression model with the BPRS value as dependent varibale and week + treatment as independent variables (explanatory variables).
# create a regression model bprs_l_reg
bprs_l_reg <- lm(bprs ~ week + treatment, data = bprs_l)
# print out a summary of the model
summary(bprs_l_reg)##
## Call:
## lm(formula = bprs ~ week + treatment, data = bprs_l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16This linear regression gives as a results that the measurement week has a significant effect on the BPRS values. With each new BRPS determination every week the BRPS value decreases by 2.2 units. Treatment 2 could have a better effect on the BPRS values of the subjects, but it does not show a significant influence. But are the treatments really helping the patients and are there some effects on the model by the subjects (the patients) with influence the model - therefore we add the subjects as a random effect term into a random intercept model.
4.3. Random Intercept Model
Now we perform a random intercept model on the BPRS data. We add as explanatory variables the measurement periods “week” and the “treatment” and add as the random-effects term the subjects (the 40 men which udnergoing the treatment and measurements). This model will consider the linear regression fit for each subject to the other subjects.
# Create a random intercept model
bprs_l_rim <- lmer(bprs ~ week + treatment + (1 | subject), data = bprs_l, REML = FALSE)
# Print the summary of the model
summary(bprs_l_rim)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: bprs_l
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9090 24.334
## week -2.2704 0.2084 -10.896
## treatment2 0.5722 1.0761 0.532
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000We see in the random intercept model summary that the standard error of the week (the measurement periods) is smaller compared to the standard error in the linear regression model (0.2084 compared to 0.2524) and also the standard error of the treatment 2 is smaller (1.0761 compared to 1.3034). That means that the subject effect was taken into account in this model. The t-value gives a value if the confidence interval of the BPRS values can be 0 - here that is not the case. Generally: the mixed model (lmer) takes into account the correlations between the observations (here between the subjects), so it is more accurate compared to the linear regression model. The p-value can be calculated by inverse function of the t-values (a function to invert the t value to the p value).
4.4. Random Intercept and Random Slope Model
Here we are fitting a random intercept and random slope model to the BPRS values. That allows the linear regression to fit for each subject (patient) to differ in the intercept and in the slope. So it will be possible to take the subject differences into accountof the BPRS values and also in the effect of the measurement period. So in the next model the “week” and the “subject” is included as a random-effect term.
# create a random intercept and random slope model
bprs_l_rim1 <- lmer(bprs ~ week + treatment + (week | subject), data = bprs_l, REML = FALSE)
# here the other times (weeks) are included - you get a regression model defined for each subject for each time period (for each week) - so here 9 (0-8 weeks) different regression models
# print a summary of the model
summary(bprs_l_rim1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: bprs_l
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7977
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8222 8.0512
## week 0.9609 0.9803 -0.51
## Residual 97.4304 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 2.1052 22.066
## week -2.2704 0.2977 -7.626
## treatment2 0.5722 1.0405 0.550
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000The model results show not so much difference compared to the model before. To check difference in the models we perform an ANOVA.
# perform an ANOVA test on the two models
anova(bprs_l_rim1, bprs_l_rim)## Data: bprs_l
## Models:
## bprs_l_rim: bprs ~ week + treatment + (1 | subject)
## bprs_l_rim1: bprs ~ week + treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## bprs_l_rim 5 2748.7 2768.1 -1369.4 2738.7
## bprs_l_rim1 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The ANOVA shows that the “Random Intercept and Random Slope Model” (taking week and subject into account) gives a better fit for the data (p-value of 0.02) with a significant outcome.
4.5. Random Intercept and Random Slope Model with interaction
Now we perform a “Random Intercept and Random Slope Model” where we include an interaction of the treatment and the time (weeks) to see if the treatment time period interacts with the BRPS value outcome.
# create a random intercept and random slope model
bprs_l_rim2 <- lmer(bprs ~ week * treatment + (week | subject), data = bprs_l, REML = FALSE)
# print a summary of the model
summary(bprs_l_rim2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week * treatment + (week | subject)
## Data: bprs_l
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0768 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.9964 8.0620
## week 0.9687 0.9842 -0.51
## Residual 96.4707 9.8220
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.8856 2.2521 21.262
## week -2.6283 0.3589 -7.323
## treatment2 -2.2911 1.9090 -1.200
## week:treatment2 0.7158 0.4010 1.785
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840# perform an ANOVA test on the two models
anova(bprs_l_rim2, bprs_l_rim1)## Data: bprs_l
## Models:
## bprs_l_rim1: bprs ~ week + treatment + (week | subject)
## bprs_l_rim2: bprs ~ week * treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## bprs_l_rim1 7 2745.4 2772.6 -1365.7 2731.4
## bprs_l_rim2 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The ANOVA shows that the “Random Intercept and Random Slope Model” with the time * treatment interaction fits likely better on the data (p-value of 0.075 < 0.1). So there is an interaction of the treatments with the treament time.
Now, we check the BRPS data again on a graph, then we add fitted BPRS values created by the “Random Intercept and Random Slope Model” with the time * treatment interaction to the data table and draw the plot with the fitted BPRS values.
# draw the plot of bprs_l
ggplot(bprs_l, aes(x = week, y = bprs, linetype = subject)) +
geom_line(color = "darkgreen") +
scale_linetype_manual(values = rep(1:6, times = 4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_x_continuous(name = "Weeks", breaks = seq(0,8,1)) +
scale_y_continuous(name = "BPRS values observed", breaks = seq(10,100,5)) +
theme(legend.position = "bottom") +
ggtitle("BPRS: original values by treatment") +
theme(legend.box.background = element_rect(),legend.box.margin = margin(2, 2, 2, 2))
Now we calculate fitted BPRS values according to the latest model and add the fitted BRPS values to the data table. These fitted values are then plotted. The plot shows nicely the BPRS alue development according to the model.
# Create a vector of the fitted values
bprs_l_fit <- fitted(bprs_l_rim2)
# Create a new column "fitted_bprs" to bprs_l
bprs_l$fitted_bprs <- round(bprs_l_fit, digits = 2)
# Check the BRPS data table again
bprs_l %>% knitr::kable() %>% kable_styling(bootstrap_options = "striped", full_width = FALSE, position = "center") %>% scroll_box(height = "300px")| treatment | subject | weeks | bprs | week | fitted_bprs |
|---|---|---|---|---|---|
| 1 | 1 | week0 | 42 | 0 | 49.24 |
| 1 | 2 | week0 | 58 | 0 | 46.97 |
| 1 | 3 | week0 | 54 | 0 | 47.66 |
| 1 | 4 | week0 | 55 | 0 | 49.85 |
| 1 | 5 | week0 | 72 | 0 | 66.39 |
| 1 | 6 | week0 | 48 | 0 | 42.59 |
| 1 | 7 | week0 | 71 | 0 | 54.58 |
| 1 | 8 | week0 | 30 | 0 | 47.93 |
| 1 | 9 | week0 | 41 | 0 | 42.08 |
| 1 | 10 | week0 | 57 | 0 | 53.35 |
| 1 | 11 | week0 | 30 | 0 | 59.97 |
| 1 | 12 | week0 | 55 | 0 | 48.42 |
| 1 | 13 | week0 | 36 | 0 | 50.36 |
| 1 | 14 | week0 | 38 | 0 | 40.93 |
| 1 | 15 | week0 | 66 | 0 | 55.07 |
| 1 | 16 | week0 | 41 | 0 | 44.43 |
| 1 | 17 | week0 | 45 | 0 | 43.93 |
| 1 | 18 | week0 | 39 | 0 | 36.82 |
| 1 | 19 | week0 | 24 | 0 | 38.14 |
| 1 | 20 | week0 | 38 | 0 | 39.01 |
| 2 | 1 | week0 | 52 | 0 | 46.95 |
| 2 | 2 | week0 | 30 | 0 | 44.68 |
| 2 | 3 | week0 | 65 | 0 | 45.36 |
| 2 | 4 | week0 | 37 | 0 | 47.56 |
| 2 | 5 | week0 | 59 | 0 | 64.10 |
| 2 | 6 | week0 | 30 | 0 | 40.30 |
| 2 | 7 | week0 | 69 | 0 | 52.29 |
| 2 | 8 | week0 | 62 | 0 | 45.64 |
| 2 | 9 | week0 | 38 | 0 | 39.79 |
| 2 | 10 | week0 | 65 | 0 | 51.05 |
| 2 | 11 | week0 | 78 | 0 | 57.68 |
| 2 | 12 | week0 | 38 | 0 | 46.13 |
| 2 | 13 | week0 | 63 | 0 | 48.07 |
| 2 | 14 | week0 | 40 | 0 | 38.64 |
| 2 | 15 | week0 | 40 | 0 | 52.78 |
| 2 | 16 | week0 | 54 | 0 | 42.14 |
| 2 | 17 | week0 | 33 | 0 | 41.63 |
| 2 | 18 | week0 | 28 | 0 | 34.53 |
| 2 | 19 | week0 | 52 | 0 | 35.85 |
| 2 | 20 | week0 | 47 | 0 | 36.71 |
| 1 | 1 | week1 | 36 | 1 | 47.97 |
| 1 | 2 | week1 | 68 | 1 | 43.72 |
| 1 | 3 | week1 | 55 | 1 | 44.47 |
| 1 | 4 | week1 | 77 | 1 | 47.41 |
| 1 | 5 | week1 | 75 | 1 | 62.11 |
| 1 | 6 | week1 | 43 | 1 | 40.04 |
| 1 | 7 | week1 | 61 | 1 | 51.11 |
| 1 | 8 | week1 | 36 | 1 | 46.36 |
| 1 | 9 | week1 | 43 | 1 | 39.11 |
| 1 | 10 | week1 | 51 | 1 | 50.79 |
| 1 | 11 | week1 | 34 | 1 | 58.01 |
| 1 | 12 | week1 | 52 | 1 | 45.55 |
| 1 | 13 | week1 | 32 | 1 | 47.26 |
| 1 | 14 | week1 | 35 | 1 | 38.73 |
| 1 | 15 | week1 | 68 | 1 | 51.69 |
| 1 | 16 | week1 | 35 | 1 | 41.50 |
| 1 | 17 | week1 | 38 | 1 | 42.30 |
| 1 | 18 | week1 | 35 | 1 | 35.03 |
| 1 | 19 | week1 | 28 | 1 | 35.66 |
| 1 | 20 | week1 | 34 | 1 | 36.35 |
| 2 | 1 | week1 | 73 | 1 | 46.39 |
| 2 | 2 | week1 | 23 | 1 | 42.14 |
| 2 | 3 | week1 | 31 | 1 | 42.89 |
| 2 | 4 | week1 | 31 | 1 | 45.83 |
| 2 | 5 | week1 | 67 | 1 | 60.53 |
| 2 | 6 | week1 | 33 | 1 | 38.47 |
| 2 | 7 | week1 | 52 | 1 | 49.53 |
| 2 | 8 | week1 | 54 | 1 | 44.79 |
| 2 | 9 | week1 | 40 | 1 | 37.53 |
| 2 | 10 | week1 | 44 | 1 | 49.21 |
| 2 | 11 | week1 | 95 | 1 | 56.44 |
| 2 | 12 | week1 | 41 | 1 | 43.97 |
| 2 | 13 | week1 | 65 | 1 | 45.68 |
| 2 | 14 | week1 | 37 | 1 | 37.16 |
| 2 | 15 | week1 | 36 | 1 | 50.11 |
| 2 | 16 | week1 | 45 | 1 | 39.92 |
| 2 | 17 | week1 | 41 | 1 | 40.72 |
| 2 | 18 | week1 | 30 | 1 | 33.45 |
| 2 | 19 | week1 | 43 | 1 | 34.09 |
| 2 | 20 | week1 | 36 | 1 | 34.78 |
| 1 | 1 | week2 | 36 | 2 | 46.69 |
| 1 | 2 | week2 | 61 | 2 | 40.46 |
| 1 | 3 | week2 | 41 | 2 | 41.28 |
| 1 | 4 | week2 | 49 | 2 | 44.96 |
| 1 | 5 | week2 | 72 | 2 | 57.82 |
| 1 | 6 | week2 | 41 | 2 | 37.49 |
| 1 | 7 | week2 | 47 | 2 | 47.64 |
| 1 | 8 | week2 | 38 | 2 | 44.79 |
| 1 | 9 | week2 | 39 | 2 | 36.14 |
| 1 | 10 | week2 | 51 | 2 | 48.23 |
| 1 | 11 | week2 | 34 | 2 | 56.06 |
| 1 | 12 | week2 | 49 | 2 | 42.68 |
| 1 | 13 | week2 | 36 | 2 | 44.15 |
| 1 | 14 | week2 | 36 | 2 | 36.53 |
| 1 | 15 | week2 | 65 | 2 | 48.30 |
| 1 | 16 | week2 | 45 | 2 | 38.56 |
| 1 | 17 | week2 | 46 | 2 | 40.67 |
| 1 | 18 | week2 | 27 | 2 | 33.24 |
| 1 | 19 | week2 | 31 | 2 | 33.19 |
| 1 | 20 | week2 | 27 | 2 | 33.70 |
| 2 | 1 | week2 | 42 | 2 | 45.83 |
| 2 | 2 | week2 | 32 | 2 | 39.60 |
| 2 | 3 | week2 | 33 | 2 | 40.42 |
| 2 | 4 | week2 | 27 | 2 | 44.10 |
| 2 | 5 | week2 | 58 | 2 | 56.96 |
| 2 | 6 | week2 | 37 | 2 | 36.63 |
| 2 | 7 | week2 | 41 | 2 | 46.78 |
| 2 | 8 | week2 | 49 | 2 | 43.93 |
| 2 | 9 | week2 | 38 | 2 | 35.28 |
| 2 | 10 | week2 | 31 | 2 | 47.37 |
| 2 | 11 | week2 | 75 | 2 | 55.20 |
| 2 | 12 | week2 | 36 | 2 | 41.82 |
| 2 | 13 | week2 | 60 | 2 | 43.29 |
| 2 | 14 | week2 | 31 | 2 | 35.67 |
| 2 | 15 | week2 | 55 | 2 | 47.44 |
| 2 | 16 | week2 | 35 | 2 | 37.70 |
| 2 | 17 | week2 | 30 | 2 | 39.81 |
| 2 | 18 | week2 | 29 | 2 | 32.38 |
| 2 | 19 | week2 | 26 | 2 | 32.33 |
| 2 | 20 | week2 | 32 | 2 | 32.84 |
| 1 | 1 | week3 | 43 | 3 | 45.42 |
| 1 | 2 | week3 | 55 | 3 | 37.20 |
| 1 | 3 | week3 | 38 | 3 | 38.08 |
| 1 | 4 | week3 | 54 | 3 | 42.52 |
| 1 | 5 | week3 | 65 | 3 | 53.54 |
| 1 | 6 | week3 | 38 | 3 | 34.94 |
| 1 | 7 | week3 | 30 | 3 | 44.17 |
| 1 | 8 | week3 | 38 | 3 | 43.22 |
| 1 | 9 | week3 | 35 | 3 | 33.17 |
| 1 | 10 | week3 | 55 | 3 | 45.68 |
| 1 | 11 | week3 | 41 | 3 | 54.11 |
| 1 | 12 | week3 | 54 | 3 | 39.81 |
| 1 | 13 | week3 | 31 | 3 | 41.04 |
| 1 | 14 | week3 | 34 | 3 | 34.33 |
| 1 | 15 | week3 | 49 | 3 | 44.92 |
| 1 | 16 | week3 | 42 | 3 | 35.63 |
| 1 | 17 | week3 | 38 | 3 | 39.04 |
| 1 | 18 | week3 | 25 | 3 | 31.46 |
| 1 | 19 | week3 | 28 | 3 | 30.71 |
| 1 | 20 | week3 | 25 | 3 | 31.04 |
| 2 | 1 | week3 | 41 | 3 | 45.27 |
| 2 | 2 | week3 | 24 | 3 | 37.06 |
| 2 | 3 | week3 | 28 | 3 | 37.94 |
| 2 | 4 | week3 | 31 | 3 | 42.37 |
| 2 | 5 | week3 | 61 | 3 | 53.39 |
| 2 | 6 | week3 | 33 | 3 | 34.79 |
| 2 | 7 | week3 | 33 | 3 | 44.02 |
| 2 | 8 | week3 | 39 | 3 | 43.08 |
| 2 | 9 | week3 | 27 | 3 | 33.02 |
| 2 | 10 | week3 | 34 | 3 | 45.53 |
| 2 | 11 | week3 | 76 | 3 | 53.96 |
| 2 | 12 | week3 | 27 | 3 | 39.66 |
| 2 | 13 | week3 | 53 | 3 | 40.90 |
| 2 | 14 | week3 | 38 | 3 | 34.18 |
| 2 | 15 | week3 | 55 | 3 | 44.78 |
| 2 | 16 | week3 | 27 | 3 | 35.48 |
| 2 | 17 | week3 | 32 | 3 | 38.89 |
| 2 | 18 | week3 | 33 | 3 | 31.31 |
| 2 | 19 | week3 | 27 | 3 | 30.57 |
| 2 | 20 | week3 | 29 | 3 | 30.90 |
| 1 | 1 | week4 | 41 | 4 | 44.14 |
| 1 | 2 | week4 | 43 | 4 | 33.95 |
| 1 | 3 | week4 | 43 | 4 | 34.89 |
| 1 | 4 | week4 | 56 | 4 | 40.07 |
| 1 | 5 | week4 | 50 | 4 | 49.25 |
| 1 | 6 | week4 | 36 | 4 | 32.38 |
| 1 | 7 | week4 | 27 | 4 | 40.69 |
| 1 | 8 | week4 | 31 | 4 | 41.65 |
| 1 | 9 | week4 | 28 | 4 | 30.19 |
| 1 | 10 | week4 | 53 | 4 | 43.12 |
| 1 | 11 | week4 | 36 | 4 | 52.15 |
| 1 | 12 | week4 | 48 | 4 | 36.94 |
| 1 | 13 | week4 | 25 | 4 | 37.94 |
| 1 | 14 | week4 | 25 | 4 | 32.13 |
| 1 | 15 | week4 | 36 | 4 | 41.54 |
| 1 | 16 | week4 | 31 | 4 | 32.69 |
| 1 | 17 | week4 | 40 | 4 | 37.41 |
| 1 | 18 | week4 | 29 | 4 | 29.67 |
| 1 | 19 | week4 | 29 | 4 | 28.23 |
| 1 | 20 | week4 | 25 | 4 | 28.39 |
| 2 | 1 | week4 | 39 | 4 | 44.72 |
| 2 | 2 | week4 | 20 | 4 | 34.52 |
| 2 | 3 | week4 | 22 | 4 | 35.47 |
| 2 | 4 | week4 | 31 | 4 | 40.65 |
| 2 | 5 | week4 | 49 | 4 | 49.83 |
| 2 | 6 | week4 | 28 | 4 | 32.95 |
| 2 | 7 | week4 | 34 | 4 | 41.27 |
| 2 | 8 | week4 | 55 | 4 | 42.23 |
| 2 | 9 | week4 | 31 | 4 | 30.77 |
| 2 | 10 | week4 | 39 | 4 | 43.70 |
| 2 | 11 | week4 | 66 | 4 | 52.72 |
| 2 | 12 | week4 | 29 | 4 | 37.51 |
| 2 | 13 | week4 | 52 | 4 | 38.51 |
| 2 | 14 | week4 | 35 | 4 | 32.70 |
| 2 | 15 | week4 | 42 | 4 | 42.11 |
| 2 | 16 | week4 | 25 | 4 | 33.26 |
| 2 | 17 | week4 | 46 | 4 | 37.98 |
| 2 | 18 | week4 | 30 | 4 | 30.24 |
| 2 | 19 | week4 | 24 | 4 | 28.81 |
| 2 | 20 | week4 | 25 | 4 | 28.96 |
| 1 | 1 | week5 | 40 | 5 | 42.87 |
| 1 | 2 | week5 | 34 | 5 | 30.69 |
| 1 | 3 | week5 | 28 | 5 | 31.70 |
| 1 | 4 | week5 | 50 | 5 | 37.63 |
| 1 | 5 | week5 | 39 | 5 | 44.97 |
| 1 | 6 | week5 | 29 | 5 | 29.83 |
| 1 | 7 | week5 | 40 | 5 | 37.22 |
| 1 | 8 | week5 | 26 | 5 | 40.08 |
| 1 | 9 | week5 | 22 | 5 | 27.22 |
| 1 | 10 | week5 | 43 | 5 | 40.57 |
| 1 | 11 | week5 | 36 | 5 | 50.20 |
| 1 | 12 | week5 | 43 | 5 | 34.06 |
| 1 | 13 | week5 | 25 | 5 | 34.83 |
| 1 | 14 | week5 | 27 | 5 | 29.92 |
| 1 | 15 | week5 | 32 | 5 | 38.16 |
| 1 | 16 | week5 | 31 | 5 | 29.76 |
| 1 | 17 | week5 | 33 | 5 | 35.78 |
| 1 | 18 | week5 | 28 | 5 | 27.88 |
| 1 | 19 | week5 | 21 | 5 | 25.76 |
| 1 | 20 | week5 | 27 | 5 | 25.73 |
| 2 | 1 | week5 | 38 | 5 | 44.16 |
| 2 | 2 | week5 | 20 | 5 | 31.98 |
| 2 | 3 | week5 | 25 | 5 | 32.99 |
| 2 | 4 | week5 | 26 | 5 | 38.92 |
| 2 | 5 | week5 | 38 | 5 | 46.26 |
| 2 | 6 | week5 | 26 | 5 | 31.12 |
| 2 | 7 | week5 | 37 | 5 | 38.51 |
| 2 | 8 | week5 | 51 | 5 | 41.37 |
| 2 | 9 | week5 | 24 | 5 | 28.51 |
| 2 | 10 | week5 | 34 | 5 | 41.86 |
| 2 | 11 | week5 | 64 | 5 | 51.49 |
| 2 | 12 | week5 | 27 | 5 | 35.35 |
| 2 | 13 | week5 | 32 | 5 | 36.12 |
| 2 | 14 | week5 | 30 | 5 | 31.21 |
| 2 | 15 | week5 | 30 | 5 | 39.45 |
| 2 | 16 | week5 | 22 | 5 | 31.04 |
| 2 | 17 | week5 | 43 | 5 | 37.07 |
| 2 | 18 | week5 | 26 | 5 | 29.17 |
| 2 | 19 | week5 | 32 | 5 | 27.05 |
| 2 | 20 | week5 | 23 | 5 | 27.02 |
| 1 | 1 | week6 | 38 | 6 | 41.60 |
| 1 | 2 | week6 | 28 | 6 | 27.43 |
| 1 | 3 | week6 | 29 | 6 | 28.51 |
| 1 | 4 | week6 | 47 | 6 | 35.18 |
| 1 | 5 | week6 | 32 | 6 | 40.69 |
| 1 | 6 | week6 | 33 | 6 | 27.28 |
| 1 | 7 | week6 | 30 | 6 | 33.75 |
| 1 | 8 | week6 | 26 | 6 | 38.51 |
| 1 | 9 | week6 | 20 | 6 | 24.25 |
| 1 | 10 | week6 | 43 | 6 | 38.01 |
| 1 | 11 | week6 | 38 | 6 | 48.24 |
| 1 | 12 | week6 | 37 | 6 | 31.19 |
| 1 | 13 | week6 | 21 | 6 | 31.73 |
| 1 | 14 | week6 | 25 | 6 | 27.72 |
| 1 | 15 | week6 | 27 | 6 | 34.78 |
| 1 | 16 | week6 | 29 | 6 | 26.82 |
| 1 | 17 | week6 | 27 | 6 | 34.15 |
| 1 | 18 | week6 | 21 | 6 | 26.09 |
| 1 | 19 | week6 | 22 | 6 | 23.28 |
| 1 | 20 | week6 | 21 | 6 | 23.08 |
| 2 | 1 | week6 | 43 | 6 | 43.60 |
| 2 | 2 | week6 | 19 | 6 | 29.44 |
| 2 | 3 | week6 | 24 | 6 | 30.52 |
| 2 | 4 | week6 | 24 | 6 | 37.19 |
| 2 | 5 | week6 | 37 | 6 | 42.69 |
| 2 | 6 | week6 | 27 | 6 | 29.28 |
| 2 | 7 | week6 | 37 | 6 | 35.76 |
| 2 | 8 | week6 | 55 | 6 | 40.52 |
| 2 | 9 | week6 | 22 | 6 | 26.25 |
| 2 | 10 | week6 | 41 | 6 | 40.02 |
| 2 | 11 | week6 | 64 | 6 | 50.25 |
| 2 | 12 | week6 | 21 | 6 | 33.20 |
| 2 | 13 | week6 | 37 | 6 | 33.73 |
| 2 | 14 | week6 | 33 | 6 | 29.73 |
| 2 | 15 | week6 | 26 | 6 | 36.78 |
| 2 | 16 | week6 | 22 | 6 | 28.82 |
| 2 | 17 | week6 | 43 | 6 | 36.15 |
| 2 | 18 | week6 | 36 | 6 | 28.10 |
| 2 | 19 | week6 | 21 | 6 | 25.29 |
| 2 | 20 | week6 | 23 | 6 | 25.08 |
| 1 | 1 | week7 | 47 | 7 | 40.32 |
| 1 | 2 | week7 | 28 | 7 | 24.18 |
| 1 | 3 | week7 | 25 | 7 | 25.32 |
| 1 | 4 | week7 | 42 | 7 | 32.74 |
| 1 | 5 | week7 | 38 | 7 | 36.40 |
| 1 | 6 | week7 | 27 | 7 | 24.72 |
| 1 | 7 | week7 | 31 | 7 | 30.28 |
| 1 | 8 | week7 | 25 | 7 | 36.94 |
| 1 | 9 | week7 | 23 | 7 | 21.28 |
| 1 | 10 | week7 | 39 | 7 | 35.46 |
| 1 | 11 | week7 | 36 | 7 | 46.29 |
| 1 | 12 | week7 | 36 | 7 | 28.32 |
| 1 | 13 | week7 | 19 | 7 | 28.62 |
| 1 | 14 | week7 | 26 | 7 | 25.52 |
| 1 | 15 | week7 | 30 | 7 | 31.40 |
| 1 | 16 | week7 | 26 | 7 | 23.88 |
| 1 | 17 | week7 | 31 | 7 | 32.52 |
| 1 | 18 | week7 | 25 | 7 | 24.31 |
| 1 | 19 | week7 | 23 | 7 | 20.81 |
| 1 | 20 | week7 | 19 | 7 | 20.43 |
| 2 | 1 | week7 | 62 | 7 | 43.04 |
| 2 | 2 | week7 | 18 | 7 | 26.90 |
| 2 | 3 | week7 | 31 | 7 | 28.04 |
| 2 | 4 | week7 | 26 | 7 | 35.46 |
| 2 | 5 | week7 | 36 | 7 | 39.12 |
| 2 | 6 | week7 | 23 | 7 | 27.44 |
| 2 | 7 | week7 | 38 | 7 | 33.00 |
| 2 | 8 | week7 | 59 | 7 | 39.66 |
| 2 | 9 | week7 | 21 | 7 | 24.00 |
| 2 | 10 | week7 | 42 | 7 | 38.18 |
| 2 | 11 | week7 | 60 | 7 | 49.01 |
| 2 | 12 | week7 | 22 | 7 | 31.04 |
| 2 | 13 | week7 | 52 | 7 | 31.34 |
| 2 | 14 | week7 | 30 | 7 | 28.24 |
| 2 | 15 | week7 | 30 | 7 | 34.12 |
| 2 | 16 | week7 | 22 | 7 | 26.60 |
| 2 | 17 | week7 | 43 | 7 | 35.24 |
| 2 | 18 | week7 | 33 | 7 | 27.03 |
| 2 | 19 | week7 | 21 | 7 | 23.53 |
| 2 | 20 | week7 | 23 | 7 | 23.15 |
| 1 | 1 | week8 | 51 | 8 | 39.05 |
| 1 | 2 | week8 | 28 | 8 | 20.92 |
| 1 | 3 | week8 | 24 | 8 | 22.13 |
| 1 | 4 | week8 | 46 | 8 | 30.29 |
| 1 | 5 | week8 | 32 | 8 | 32.12 |
| 1 | 6 | week8 | 25 | 8 | 22.17 |
| 1 | 7 | week8 | 31 | 8 | 26.81 |
| 1 | 8 | week8 | 24 | 8 | 35.37 |
| 1 | 9 | week8 | 21 | 8 | 18.31 |
| 1 | 10 | week8 | 32 | 8 | 32.90 |
| 1 | 11 | week8 | 36 | 8 | 44.33 |
| 1 | 12 | week8 | 31 | 8 | 25.45 |
| 1 | 13 | week8 | 22 | 8 | 25.52 |
| 1 | 14 | week8 | 26 | 8 | 23.32 |
| 1 | 15 | week8 | 37 | 8 | 28.02 |
| 1 | 16 | week8 | 30 | 8 | 20.95 |
| 1 | 17 | week8 | 27 | 8 | 30.89 |
| 1 | 18 | week8 | 20 | 8 | 22.52 |
| 1 | 19 | week8 | 22 | 8 | 18.33 |
| 1 | 20 | week8 | 21 | 8 | 17.77 |
| 2 | 1 | week8 | 50 | 8 | 42.48 |
| 2 | 2 | week8 | 20 | 8 | 24.35 |
| 2 | 3 | week8 | 32 | 8 | 25.57 |
| 2 | 4 | week8 | 23 | 8 | 33.73 |
| 2 | 5 | week8 | 35 | 8 | 35.55 |
| 2 | 6 | week8 | 21 | 8 | 25.61 |
| 2 | 7 | week8 | 35 | 8 | 30.25 |
| 2 | 8 | week8 | 66 | 8 | 38.81 |
| 2 | 9 | week8 | 21 | 8 | 21.74 |
| 2 | 10 | week8 | 39 | 8 | 36.34 |
| 2 | 11 | week8 | 75 | 8 | 47.77 |
| 2 | 12 | week8 | 23 | 8 | 28.89 |
| 2 | 13 | week8 | 28 | 8 | 28.95 |
| 2 | 14 | week8 | 27 | 8 | 26.75 |
| 2 | 15 | week8 | 37 | 8 | 31.45 |
| 2 | 16 | week8 | 22 | 8 | 24.38 |
| 2 | 17 | week8 | 43 | 8 | 34.33 |
| 2 | 18 | week8 | 30 | 8 | 25.96 |
| 2 | 19 | week8 | 21 | 8 | 21.77 |
| 2 | 20 | week8 | 23 | 8 | 21.21 |
# Draw the plot of fitted bprs values
ggplot(bprs_l, aes(x = week, y = fitted_bprs, linetype = subject)) +
geom_line(color = "darkred") +
scale_linetype_manual(values = rep(1:6, times = 4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_x_continuous(name = "Weeks", breaks = seq(0,8,1)) +
scale_y_continuous(name = "BPRS values modeled", breaks = seq(10,100,5)) +
theme(legend.position = "bottom") +
ggtitle("BPRS: fitted values by treatment") +
theme(legend.box.background = element_rect(),legend.box.margin = margin(2, 2, 2, 2))